Hermitian Dickson Dualities for Codes over Near-Fields

Abstract

A Dickson near-field is obtained from ${\mathbb{F}}_{p^2}$ by twisting multiplication so that distributivity holds only on the right. In this work, we develop a basic theory of right-linear codes of length n over $\mathrm{NF}\left(p^2\right)$. We show that every right-linear code is right-monomially equivalent to a code with a systematic generator matrix, obtained via one-sided row operations. Using Galois conjugation, we introduce the Hermitian Dickson inner product and define the associated dual code, giving an explicit parity-check description in the ${\mathbb{F}}_p$-systematic case. We also provide effective criteria for Hermitian Dickson LCD, self-orthogonal, and self-dual codes, and we classify Hermitian Dickson self-orthogonal codes in short lengths.

1. Introduction

Since Shannon’s foundational work in 1948, linear codes have been studied primarily over finite fields, where the ambient space is a vector space and duality is governed by a bilinear or Hermitian inner product; classical references include [1,2]. A major expansion of the subject began in the 1990s, when it was recognized that several families of highly structured nonlinear binary codes possess a natural linear algebraic description over ${\mathbb{Z}}_4$, thereby explaining their formal duality and related enumerator identities [3]. This insight helped establish codes over rings as a robust generalization of the field case; see, e.g., [4].

In parallel, there has been renewed interest in codes over non-unital rings, where duality and hull phenomena acquire new features and motivate refined algebraic criteria; see, for instance, [5,6,7,8]. Another natural direction is to weaken distributivity rather than commutativity or the existence of a unit. Near-rings and near-fields provide precisely such a setting: one retains an additive group together with an associative multiplication that is distributive on only one side, and in the near-field case, each nonzero element admits a multiplicative inverse.

Among these, the Dickson near-fields form a distinguished and historically important family originating in Dickson’s work on finite algebras [9]; see also the classical treatment of finite near-fields in [10]. From the standpoint of coding theory, this one-sided distributive environment forces an inherently asymmetric linear algebra: right scalar actions behave well, while left multiplication can fail to be additive, so standard constructions from the field case must be reformulated carefully. In particular, duality and hull phenomena—which underpin structural characterizations of LCD, self-orthogonal, and self-dual codes—are no longer automatic, since the dual of a right-linear code need not be right-linear a priori and parity-check-type descriptions may require additional hypotheses. This motivates a systematic development of Hermitian-type duality for right-linear codes over Dickson near-fields. Related one-sided linear-algebra phenomena also appear in broader near-field/near-vector-space settings; see, for instance, [11]. The starting point for coding over Dickson near-fields is the recent case study of the smallest example, $\mathrm{NF}\left(9\right)$, developed in [12].

This paper extends the study from $\mathrm{NF}\left(9\right)$ to the general Dickson near-field $\mathrm{NF}\left(p^2\right)$ associated with an odd prime p. This extension is significant because it moves from a single small case to an infinite family of genuinely one-sided distributive algebras, allowing us to separate features that are intrinsic to near-field linear algebra from phenomena that may be accidental in $\mathrm{NF}\left(9\right)$. In contrast to the field case, $\mathrm{NF}{\left(p^2\right)}^n$ is not a two-sided vector space: left distributivity may fail in $\mathrm{NF}\left(p^2\right)$. Hence, even if $\mathcal{C}$ is right-linear, it is not automatic that its Hermitian dual ${\mathcal{C}}^{{\perp }_H}$ is again right-linear; this motivates the stability hypothesis used later to recover right-linearity of the dual. One of the main theoretical hurdles is therefore to identify natural hypotheses under which Hermitian duality behaves well in the right-linear category and admits parity-check-type descriptions.

We develop a self-contained framework for right-linear codes in $\mathrm{NF}{\left(p^2\right)}^n$ tailored to this one-sided setting. We formalize generator-matrix methods and right monomial equivalence and show that every right-linear code is right-monomially equivalent to one with a systematic generator matrix obtained via one-sided row operations. Using Galois (Frobenius) conjugation (the Frobenius automorphism of ${\mathbb{F}}_{p^2}$), we introduce the Hermitian Dickson inner product and the associated duality. A key novelty is our treatment of the Hermitian right dual: under a natural left-stability hypothesis on the conjugate code, we prove that ${\mathcal{C}}^{{\perp }_H}$ is again right-linear and contains the right row span of an explicit matrix $H=[-{\overline{A}}^{\top }I]$, providing an effective substitute for classical parity-check matrices in this non-field setting. In the ${\mathbb{F}}_p$-systematic regime we derive explicit descriptions together with the expected dimension relation. We then establish Gram-type criteria, in terms of $\Gamma =I_k+A{A}^{\top }$, characterizing Hermitian Dickson LCD, self-orthogonal, and self-dual codes. Finally, we provide computational data by classifying Hermitian Dickson self-orthogonal right-linear codes in short lengths. As an additional perspective, one-sided/asymmetric mechanisms also appear in modern optimization and learning through nonlocal or tempered memory update rules; see, for example, Tempered Fractional Gradient Descent in [13].

The paper is organized as follows. Section 2 recalls the Dickson near-field structure on ${\mathbb{F}}_{p^2}$ and introduces the one-sided linear algebra used throughout. Section 3 develops the basic theory of right-linear codes over $\mathrm{NF}\left(p^2\right)$, including systematic form and parameter bounds. Section 4 introduces the Hermitian Dickson inner product and establishes its fundamental properties and studies the resulting duality, with emphasis on the systematic case and Gram-type criteria. Section 5 classifies Hermitian Dickson self-orthogonal right-linear codes over $\mathrm{NF}\left(p^2\right)$ for $p\in \{3,5,7\}$, in lengths at most 8 and right dimension at most 2. Finally, Section 6 collects our conclusions and several directions for further work.

2. The Dickson Near-Field on ${\mathbb{F}}_{p^2}$

Let p be an odd prime. Fix a nonsquare $\delta \in {\mathbb{F}}_p^{\times }$ and write ${\mathbb{F}}_{p^2}=\{a+b\alpha :a,b\in {\mathbb{F}}_p,{\alpha }^2=\delta \}.$ Let $\phi :{\mathbb{F}}_{p^2}\to {\mathbb{F}}_{p^2}$ be the Frobenius automorphism, $\phi \left(t\right)=t^p$. Since ${\mathbb{F}}_{p^2}^{\times }$ is cyclic of order $p^2-1$, it has a unique subgroup of index 2, namely the set of all nonzero squares; denote it by $H_0=\{z^2:z\in {\mathbb{F}}_{p^2}^{\times }\}$.

For $y\in {\mathbb{F}}_{p^2}$, define the map ${\sigma }_y:{\mathbb{F}}_{p^2}\to {\mathbb{F}}_{p^2}$ by

$${\sigma }_y=\left\{\begin{array}{cc}\mathrm{id},\hfill & \mathrm{if}y=0\mathrm{or}y\in H_0,\hfill \\ \phi ,\hfill & \mathrm{if}y\in {\mathbb{F}}_{p^2}^{\times }\setminus H_0.\hfill \end{array}\right.$$

Following [9], define the Dickson right product on ${\mathbb{F}}_{p^2}$ by

$$x\circ y={\sigma }_y\left(x\right)y$$

(1)

where juxtaposition denotes the usual field multiplication in ${\mathbb{F}}_{p^2}$. Equivalently, for $y\ne 0$,

$$x\circ y=\left\{\begin{array}{cc}xy,\hfill & y\in H_0,\hfill \\ x^{p}y,\hfill & y\in {\mathbb{F}}_{p^2}^{\times }\setminus H_0.\hfill \end{array}\right.$$

We write $\mathrm{NF}\left(p^2\right)=({\mathbb{F}}_{p^2},+,\circ )$ for the resulting Dickson near-field of order $p^2$.

Proposition 1

([9]).  The Dickson product ∘ is associative on ${\mathbb{F}}_{p^2}$. Moreover, ${\mathbb{F}}_{p^2}^{\times }$ is a group under ∘.

Lemma 1.

For all $x,y,z\in {\mathbb{F}}_{p^2}$:

 (a) 

Right distributivity: $(x+z)\circ y=(x\circ y)+(z\circ y)$.

 (b) 

Identity: $x\circ 1=x$ and $1\circ y=y$.

 (c) 

Zero: $x\circ 0=0$ and $0\circ y=0$.

 (d) 

Inverses: If $y\ne 0$, then y has a unique two-sided inverse with respect to ∘. In particular, the unique element $u\in {\mathbb{F}}_{p^2}$ satisfying $u\circ y=1$ and $y\circ u=1$ is

$$u={\sigma }_y\left(y^{-1}\right),$$

where $y^{-1}$ denotes the inverse of y in the usual field multiplication of ${\mathbb{F}}_{p^2}$.

Proof. 

(a)

For any $y\in {\mathbb{F}}_{p^2}$,

$$(x+z)\circ y={\sigma }_y(x+z)y=({\sigma }_y\left(x\right)+{\sigma }_y\left(z\right))y={\sigma }_y\left(x\right)y+{\sigma }_y\left(z\right)y=(x\circ y)+(z\circ y),$$

since ${\sigma }_y\in \{\mathrm{id},\phi \}$ is additive.

(b)

Since $1\in H_0$, we have ${\sigma }_1=\mathrm{id}$, hence $x\circ 1={\sigma }_1\left(x\right)1=x$. Also ${\sigma }_y\left(1\right)=1$ for all y, so $1\circ y={\sigma }_y\left(1\right)y=y$.

(c)

The identity $x\circ 0=0$ holds by definition. Moreover, if $y\ne 0$ then $0\circ y={\sigma }_y\left(0\right)y=0$ and it is 0 also when $y=0$.

(d)

Assume $y\ne 0$ and set $u={\sigma }_y\left(y^{-1}\right)$, where $y^{-1}$ is the inverse of y in the field ${\mathbb{F}}_{p^2}$. Then, by (1),

$$u\circ y={\sigma }_y\left(u\right)y={\sigma }_y\left({\sigma }_y\left(y^{-1}\right)\right)y.$$

Since ${\sigma }_y\in \{\mathrm{id},\phi \}$ and ${\phi }^2=\mathrm{id}$ on ${\mathbb{F}}_{p^2}$, we have ${\sigma }_y^2=\mathrm{id}$, hence $u\circ y=y^{-1}y=1$.

By Proposition 1, ${\mathbb{F}}_{p^2}^{\times }$ is a group under ∘, so y has a unique two-sided inverse with respect to ∘. Therefore $u=y^{(-1)}$ the inverse of y in $({\mathbb{F}}_{p^2}^{\times },\circ )$ and, in particular, $y\circ u=1$. Uniqueness follows from uniqueness of inverses in a group.

□

Proposition 2.

For any odd prime p, the Dickson near-field $\mathrm{NF}\left(p^2\right)$ is not left distributive over +.

Proof. 

View $({\mathbb{F}}_{p^2},+)$ as a 2-dimensional vector space over ${\mathbb{F}}_p$. Hence, every additive subgroup (i.e., every ${\mathbb{F}}_p$-subspace) has cardinality $p^d$ with $d\in \{0,1,2\}$, so its size is 1, p, or $p^2$. On the other hand,

$$|H_0\cup \left\{0\right\}|=|H_0|+1={\displaystyle \frac{p^2-1}{2}}+1={\displaystyle \frac{p^2+1}{2}},$$

which is different from 1, p, and $p^2$ for odd p. Therefore $H_0\cup \left\{0\right\}$ is not an additive subgroup and, in particular, it is not closed under addition. Thus we may choose $u,v\in H_0$ such that $u+v\notin H_0\cup \left\{0\right\}$; in particular $u+v\ne 0$, ${\sigma }_u={\sigma }_v=\mathrm{id}$, and ${\sigma }_{u+v}=\phi$.

Pick $a\in {\mathbb{F}}_{p^2}\setminus {\mathbb{F}}_p$, so that $\phi \left(a\right)\ne a$. Then

$$(a\circ u)+(a\circ v)={\sigma }_u\left(a\right)u+{\sigma }_v\left(a\right)v=au+av=a(u+v),$$

whereas

$$a\circ (u+v)={\sigma }_{u+v}\left(a\right)(u+v)=\phi \left(a\right)(u+v).$$

If left distributivity held, we would have $a\circ (u+v)=(a\circ u)+(a\circ v)$, hence $\phi \left(a\right)(u+v)=a(u+v)$. Since $u+v\ne 0$, this implies $\phi \left(a\right)=a$, a contradiction. Therefore left distributivity fails in $\mathrm{NF}\left(p^2\right)$.   □

Corollary 1.

For any odd prime p, the Dickson near-field $\mathrm{NF}\left(p^2\right)$ is a right near-field (not a field) of order $p^2$.

Proof. 

By Proposition 1, the Dickson product ∘ is associative and ${\mathbb{F}}_{p^2}^{\times }$ is a group under ∘; by Lemma 1(a), ∘ is right distributive over +. Hence, $\mathrm{NF}\left(p^2\right)$ is a right near-field. Proposition 2 shows that left distributivity fails. Since a field must be distributive on both sides, $\mathrm{NF}\left(p^2\right)$ cannot be a field, and therefore it is a proper right near-field.   □

For $n\ge 1$ set $\mathcal{V}=\mathrm{NF}{\left(p^2\right)}^n$. We extend the Dickson product ∘ coordinate-wise by

$$(v_1,\dots ,v_n)\circ a:=(v_1\circ a,\dots ,v_n\circ a),a\circ (v_1,\dots ,v_n):=(a\circ v_1,\dots ,a\circ v_n).$$

A subset $\mathcal{C}\subseteq \mathcal{V}$ is called a right near-subspace if $(\mathcal{C},+)$ is a subgroup of $(\mathcal{V},+)$ and $\mathbf{v}\circ a\in \mathcal{C}$ for all $\mathbf{v}\in \mathcal{C}$ and all $a\in \mathrm{NF}\left(p^2\right)$.

Remark 1.

For $c\in {\mathbb{F}}_p$ and $\mathbf{v}\in \mathrm{NF}{\left(p^2\right)}^n$, the notation $c\mathbf{v}$ refers to ${\mathbb{F}}_p$–scalar multiplication in the additive group $(\mathrm{NF}{\left(p^2\right)}^n,+)$, i.e., coordinate-wise multiplication by c in the field ${\mathbb{F}}_{p^2}$. Equivalently, for $c\in {\mathbb{F}}_p$ one has $c\circ v=cv$ for all $v\in \mathrm{NF}\left(p^2\right)$, and hence $\mathbf{v}\circ c=c\mathbf{v}$ for all $\mathbf{v}\in \mathrm{NF}{\left(p^2\right)}^n$.

Lemma 2.

For each fixed $a\in \mathrm{NF}\left(p^2\right)$, the map $R_a:\mathcal{V}\to \mathcal{V}$ defined by $R_a\left(\mathbf{v}\right)=\mathbf{v}\circ a$ is ${\mathbb{F}}_p$–linear.

Proof. 

Let $\mathbf{v}=(v_1,\dots ,v_n)$ and $\mathbf{w}=(w_1,\dots ,w_n)$ in $\mathcal{V}$. Using Lemma 1(a) coordinate-wise, we obtain

$$R_a(\mathbf{v}+\mathbf{w})=\left((v_1+w_1)\circ a,\dots ,(v_n+w_n)\circ a\right)=R_a\left(\mathbf{v}\right)+R_a\left(\mathbf{w}\right).$$

Now let $c\in {\mathbb{F}}_p$. If $a=0$ then $R_a\left(c\mathbf{v}\right)=\mathbf{0}=c{R}_a\left(\mathbf{v}\right)$. If $a\ne 0$, write $x\circ a={\sigma }_a\left(x\right)a$ with ${\sigma }_a\in \{\mathrm{id},\phi \}$. Since ${\sigma }_a$ is ${\mathbb{F}}_p$–linear (in particular, ${\sigma }_a\left(c\right)=c$), for each i we have

$$\left(c{v}_i\right)\circ a={\sigma }_a\left(c{v}_i\right)a=c{\sigma }_a\left(v_i\right)a=c(v_i\circ a).$$

Hence $R_a\left(c\mathbf{v}\right)=c{R}_a\left(\mathbf{v}\right)$, and $R_a$ is ${\mathbb{F}}_p$–linear.   □

We emphasize that, unlike the right action, coordinate-wise left multiplication in $\mathrm{NF}\left(p^2\right)$ need not be additive.

Example 1.

Take $n=1$, so that $\mathcal{V}=\mathrm{NF}\left(p^2\right)$. For $a\in \mathrm{NF}\left(p^2\right)$, define $T_a:\mathcal{V}\to \mathcal{V}$ by $T_a\left(v\right)=a\circ v$. By Proposition 2, there exist $a\in {\mathbb{F}}_{p^2}\setminus {\mathbb{F}}_p$ and $u,v\in H_0$ such that $u+v\notin H_0\cup \left\{0\right\}$ and $a\circ (u+v)\ne a\circ u+a\circ v$. For this choice of $(a,u,v)$ we obtain $T_a(u+v)\ne T_a\left(u\right)+T_a\left(v\right)$, so $T_a$ is not additive. Consequently, for suitable $a\in \mathrm{NF}\left(p^2\right)$, the left action $\mathbf{v}\mapsto a\circ \mathbf{v}$ fails to be additive, and hence is not ${\mathbb{F}}_p$–linear.

Motivated by this asymmetry, we single out those right near-subspaces that are stable under left multiplication. A right near-subspace $\mathcal{C}\subseteq \mathrm{NF}{\left(p^2\right)}^n$ is called left-stable if $a\circ \mathbf{v}\in \mathcal{C}$ for all $a\in \mathrm{NF}\left(p^2\right)$ and all $\mathbf{v}\in \mathcal{C}$.

Example 2.

Fix $1\le r\le n$ and consider the coordinate subspace

$$\mathcal{C}:=\{(x_1,\dots ,x_r,0,\dots ,0)\in \mathrm{NF}{\left(p^2\right)}^n:x_1,\dots ,x_r\in \mathrm{NF}\left(p^2\right)\}.$$

Then $\mathcal{C}$ is a right near-subspace of $\mathrm{NF}{\left(p^2\right)}^n$ and is left-stable. Indeed, $\mathcal{C}$ is an additive subgroup of $\mathrm{NF}{\left(p^2\right)}^n$, it is closed under the right action since $(x_1,\dots ,x_r,0,\dots ,0)\circ a\in \mathcal{C}$ for all $a\in \mathrm{NF}\left(p^2\right)$, and it is left-stable since $a\circ (x_1,\dots ,x_r,0,\dots ,0)\in \mathcal{C}$ for all $a\in \mathrm{NF}\left(p^2\right)$.

Example 3.

Take $n=3$ and consider the additive subset

$$\mathcal{D}:=\{(x,z,x+z)\in \mathrm{NF}{\left(p^2\right)}^3:x,z\in \mathrm{NF}\left(p^2\right)\}.$$

Then $\mathcal{D}$ is a right near-subspace of $\mathrm{NF}{\left(p^2\right)}^3$, but $\mathcal{D}$ is not left-stable.

First, $\mathcal{D}$ is an additive subgroup of $\mathrm{NF}{\left(p^2\right)}^3$. Moreover, for any $b\in \mathrm{NF}\left(p^2\right)$ and any $(x,z,x+z)\in \mathcal{D}$ we have

$$(x,z,x+z)\circ b=(x\circ b,z\circ b,(x+z)\circ b)=(x\circ b,z\circ b,x\circ b+z\circ b)\in \mathcal{D},$$

where the last equality uses Lemma 1(a). Hence, $\mathcal{D}$ is a right near-subspace.

To see that $\mathcal{D}$ is not left-stable, choose $a\in {\mathbb{F}}_{p^2}\setminus {\mathbb{F}}_p$ and $u,v\in H_0$ as in Proposition 2. Then $(u,v,u+v)\in \mathcal{D}$, but

$$a\circ (u,v,u+v)=(a\circ u,a\circ v,a\circ (u+v\left)\right)\notin \mathcal{D},$$

since membership in $\mathcal{D}$ would require the third coordinate to equal the sum of the first two. Thus $\mathcal{D}$ is not left-stable.

While the Dickson product is only right distributive, nested products exhibit useful commutation phenomena when one factor lies in ${\mathbb{F}}_p$ or in the subgroup $H_0$. We record these identities in the next lemma for later use.

Lemma 3.

 (a) 

For all $x\in {\mathbb{F}}_{p^2}^{\times }$ and all $c\in H_0$, one has ${\sigma }_{cx}={\sigma }_x$.

 (b) 

For any $c\in {\mathbb{F}}_p$ and all $a,u\in \mathrm{NF}\left(p^2\right)$, one has $a\circ (c\circ u)=c\circ (a\circ u)$.

 (c) 

If $a,c\in H_0$, then for all $u\in \mathrm{NF}\left(p^2\right)$, one has $a\circ (c\circ u)=c\circ (a\circ u)$.

Proof. 

(a)

Since $H_0$ has index 2 in ${\mathbb{F}}_{p^2}^{\times }$, multiplication by any $c\in H_0$ preserves the cosets of $H_0$: for $x\in {\mathbb{F}}_{p^2}^{\times }$ we have $x\in H_0$ if and only if $cx\in H_0$. By the definition of ${\sigma }_y$, this yields ${\sigma }_{cx}={\sigma }_x$.

(b)

If $u=0$ or $c=0$, then $c\circ u=0$ and both sides are 0. Assume $u\ne 0$ and $c\ne 0$. Because $c\in {\mathbb{F}}_p$, it is fixed by both $\mathrm{id}$ and $\phi$, hence ${\sigma }_u\left(c\right)=c$ and so

$$c\circ u={\sigma }_u\left(c\right)u=cu.$$

Using associativity and the definition of ∘,

$$a\circ (c\circ u)=a\circ \left(cu\right)={\sigma }_{cu}\left(a\right)\left(cu\right).$$

Since $c\in {\mathbb{F}}_p^{\times }\subseteq H_0$, part (a) gives ${\sigma }_{cu}={\sigma }_u$, and hence

$$a\circ (c\circ u)=c{\sigma }_u\left(a\right)u.$$

On the other hand, $a\circ u={\sigma }_u\left(a\right)u$, and therefore

$$c\circ (a\circ u)=c\circ \left({\sigma }_u\left(a\right)u\right)={\sigma }_{{\sigma }_u\left(a\right)u}\left(c\right){\sigma }_u\left(a\right)u=c{\sigma }_u\left(a\right)u,$$

because ${\sigma }_{{\sigma }_u\left(a\right)u}\in \{\mathrm{id},\phi \}$ fixes $c\in {\mathbb{F}}_p$. Thus $a\circ (c\circ u)=c\circ (a\circ u)$.

(c)

If $u=0$ the claim is trivial, so assume $u\ne 0$ and set $\tau :={\sigma }_u\in \{\mathrm{id},\phi \}$. Then

$$c\circ u=\tau \left(c\right)u,a\circ u=\tau \left(a\right)u.$$

Since $a,c\in H_0$ and $\tau$ is a field automorphism, we have $\tau \left(a\right),\tau \left(c\right)\in H_0$. By part (a) applied with $x=u$, we get

$${\sigma }_{\tau \left(c\right)u}={\sigma }_u=\tau ,{\sigma }_{\tau \left(a\right)u}={\sigma }_u=\tau .$$

Hence

$$\begin{array}{cc}\hfill a\circ (c\circ u)& =a\circ \left(\tau \left(c\right)u\right)={\sigma }_{\tau \left(c\right)u}\left(a\right)\tau \left(c\right)u=\tau \left(a\right)\tau \left(c\right)u,\hfill \\ \hfill c\circ (a\circ u)& =c\circ \left(\tau \left(a\right)u\right)={\sigma }_{\tau \left(a\right)u}\left(c\right)\tau \left(a\right)u=\tau \left(c\right)\tau \left(a\right)u.\hfill \end{array}$$

Since $\tau \left(a\right)\tau \left(c\right)=\tau \left(c\right)\tau \left(a\right)$ in ${\mathbb{F}}_{p^2}$, it follows that $a\circ (c\circ u)=c\circ (a\circ u)$.

□

Galois Conjugation

To streamline the Hermitian Dickson duality developed later, we recall the Frobenius automorphism on ${\mathbb{F}}_{p^2}$ and record the identities we will use, emphasizing its interaction with the Dickson product ∘. Since the Dickson near-field $\mathrm{NF}\left(p^2\right)$ and the finite field ${\mathbb{F}}_{p^2}$ share the same underlying set (only the operations differ), this conjugation is simply the usual field conjugation and is independent of the Dickson multiplication. Over ${\mathbb{F}}_{p^2}$ this map is the usual Galois conjugation $x\mapsto x^p$; in the near-field setting we refer to it as Galois conjugation. For background on Frobenius automorphisms in finite fields, see [14]; for Hermitian inner products and Hermitian duality over ${\mathbb{F}}_{p^2}$, see [2,15].

Define Galois conjugation $\overline{\phantom{x}}:\mathrm{NF}\left(p^2\right)\to \mathrm{NF}\left(p^2\right)$ by $\overline{x}=x^p$. For $n\ge 1$, extend it coordinate-wise to $\mathrm{NF}{\left(p^2\right)}^n$ by

$$\overline{(x_1,\dots ,x_n)}:=(\overline{x_1},\dots ,\overline{x_n}).$$

This map is an ${\mathbb{F}}_p$–linear bijection on $\mathrm{NF}{\left(p^2\right)}^n$. For $\mathcal{S}\subseteq \mathrm{NF}{\left(p^2\right)}^n$, write $\overline{\mathcal{S}}:=\{\overline{\mathbf{s}}:\mathbf{s}\in \mathcal{S}\}$, and say that $\mathcal{S}$ is Galois conjugation-invariant if $\overline{\mathcal{S}}=\mathcal{S}$.

Since ${\mathbb{F}}_{p^2}^{\times }$ is cyclic and $H_0$ is its unique subgroup of index 2, the Frobenius automorphism $\phi$ preserves $H_0$. In particular, ${\sigma }_{\overline{y}}={\sigma }_y$ for all $y\in {\mathbb{F}}_{p^2}$.

Lemma 4.

For all $x,y,a\in \mathrm{NF}\left(p^2\right)$, $c\in {\mathbb{F}}_p$, $\mathbf{v}\in \mathrm{NF}{\left(p^2\right)}^n$:

 (a) 

$\overline{x+y}=\overline{x}+\overline{y}$ and $\overline{c}=c$.

 (b) 

$\overline{\overline{x}}=x$.

 (c) 

$\overline{xy}=\overline{x}\overline{y}$.

 (d) 

$\overline{x\circ y}=\overline{x}\circ \overline{y}$.

 (e) 

$\overline{\mathbf{v}\circ a}=\overline{\mathbf{v}}\circ \overline{a}$.

 (f) 

$a\circ \overline{\mathbf{v}}=\overline{\overline{a}\circ \mathbf{v}}$.

Proof. 

Parts (a)–(c) are standard properties of the Frobenius automorphism on ${\mathbb{F}}_{p^2}$.

For (d), if $y=0$ then both sides are 0. Assume $y\ne 0$. By definition, $x\circ y={\sigma }_y\left(x\right)y$, hence

$$\overline{x\circ y}=\overline{{\sigma }_y\left(x\right)}\overline{y}.$$

Since $\phi \left(H_0\right)=H_0$, we have ${\sigma }_{\overline{y}}={\sigma }_y$. If ${\sigma }_y=\mathrm{id}$ then $\overline{{\sigma }_y\left(x\right)}=\overline{x}$ and the claim follows. If ${\sigma }_y=\phi$, then $\overline{{\sigma }_y\left(x\right)}=\overline{x^p}=x$ and also ${\sigma }_{\overline{y}}\left(\overline{x}\right)=\phi \left(\overline{x}\right)={\overline{x}}^p=x$, so again $\overline{x\circ y}={\sigma }_{\overline{y}}\left(\overline{x}\right)\overline{y}=\overline{x}\circ \overline{y}$.

Parts (e) and (f) follow by applying (d) coordinate-wise and using $\overline{\overline{a}}=a$.   □

Corollary 2.

A right near-subspace $\mathcal{C}\subseteq \mathrm{NF}{\left(p^2\right)}^n$ is left-stable if and only if $\overline{\mathcal{C}}$ is left-stable.

Proof. 

Assume $\mathcal{C}$ is left-stable. Let $a\in \mathrm{NF}\left(p^2\right)$ and $\mathbf{u}\in \overline{\mathcal{C}}$, and write $\mathbf{u}=\overline{\mathbf{c}}$ with $\mathbf{c}\in \mathcal{C}$. By Lemma 4(f),

$$a\circ \mathbf{u}=a\circ \overline{\mathbf{c}}=\overline{\overline{a}\circ \mathbf{c}}.$$

Since $\overline{a}\circ \mathbf{c}\in \mathcal{C}$ by left-stability of $\mathcal{C}$, we obtain $a\circ \mathbf{u}\in \overline{\mathcal{C}}$. Hence, $\overline{\mathcal{C}}$ is left-stable.

Conversely, if $\overline{\mathcal{C}}$ is left-stable, apply the forward implication to $\overline{\mathcal{C}}$. Using $\overline{\overline{\mathcal{C}}}=\mathcal{C}$ (Lemma 4(b)), we conclude that $\mathcal{C}$ is left-stable.   □

Lemma 5.

Let $M\in \mathrm{NF}{\left(p^2\right)}^{k\times n}$, and let $\langle M\rangle$ denote the right row span of M in $\mathrm{NF}{\left(p^2\right)}^n$. Then

$$\overline{\langle M\rangle }=\langle \overline{M}\rangle .$$

Proof. 

Let ${\mathbf{m}}_1,\dots ,{\mathbf{m}}_k$ be the rows of M and let $\mathbf{x}\in \langle M\rangle$. Then $\mathbf{x}={\sum }_{i=1}^k{\mathbf{m}}_i\circ a_i$ for some $a_1,\dots ,a_k\in \mathrm{NF}\left(p^2\right)$. Using Lemma 4(a) and (e),

$$\overline{\mathbf{x}}=\sum _{i=1}^k\overline{{\mathbf{m}}_i\circ a_i}=\sum _{i=1}^k\overline{{\mathbf{m}}_i}\circ \overline{a_i}\in \langle \overline{M}\rangle .$$

Hence $\overline{\langle M\rangle }\subseteq \langle \overline{M}\rangle$. Applying conjugation again and using Lemma 4(b) gives the reverse inclusion.   □

Remark 2.

For $M\in \mathrm{NF}{\left(p^2\right)}^{k\times n}$, Lemma 5 shows that $\langle M\rangle$ is Galois conjugation-invariant if and only if $\langle M\rangle =\langle \overline{M}\rangle$.

Lemma 3 gives commutation identities for nested Dickson products involving scalars from ${\mathbb{F}}_p$ or $H_0$. The next lemma complements this by providing a structural reformulation: it expresses left-stability of $\overline{\mathcal{D}}$ as an equivalent membership condition back in $\mathcal{D}$.

Lemma 6.

Let $\mathcal{D}\subseteq \mathrm{NF}{\left(p^2\right)}^n$ be a right near-subspace. The following are equivalent:

 (i) 

$\overline{a\circ \overline{\mathbf{c}}}\in \mathcal{D}$ for all $a\in \mathrm{NF}\left(p^2\right)$ and all $\mathbf{c}\in \mathcal{D}$.

 (ii) 

$a\circ \mathbf{u}\in \overline{\mathcal{D}}$ for all $a\in \mathrm{NF}\left(p^2\right)$ and all $\mathbf{u}\in \overline{\mathcal{D}}$.

Moreover, these conditions hold if and only if $\mathcal{D}$ is left-stable.

Proof. 

(i)⇒(ii). Let $\mathbf{u}\in \overline{\mathcal{D}}$, so $\mathbf{u}=\overline{\mathbf{c}}$ for some $\mathbf{c}\in \mathcal{D}$. By (i), $\overline{a\circ \overline{\mathbf{c}}}\in \mathcal{D}$, and conjugating gives $a\circ \mathbf{u}\in \overline{\mathcal{D}}$.

(ii)⇒(i). Let $\mathbf{c}\in \mathcal{D}$ and set $\mathbf{u}=\overline{\mathbf{c}}\in \overline{\mathcal{D}}$. By (ii), $a\circ \overline{\mathbf{c}}\in \overline{\mathcal{D}}$, so $a\circ \overline{\mathbf{c}}=\overline{\mathbf{d}}$ for some $\mathbf{d}\in \mathcal{D}$. Conjugating yields $\overline{a\circ \overline{\mathbf{c}}}=\mathbf{d}\in \mathcal{D}$.

Finally, (ii) is exactly left-stability of $\overline{\mathcal{D}}$, and Corollary 2 implies that $\overline{\mathcal{D}}$ is left-stable if and only if $\mathcal{D}$ is left-stable.   □

3. Right Codes over $\mathbf{NF}\left({\mathit{p}}^{\mathbf{2}}\right)$

Throughout this section, fix an integer $n\ge 1$ and write $\mathcal{V}=\mathrm{NF}{\left(p^2\right)}^n$. All vector and matrix additions are taken in the additive group $(\mathrm{NF}\left(p^2\right),+)$, and the coordinate-wise scalar action is given by the Dickson product ∘.

3.1. Basic Definitions

A right-linear code over $\mathrm{NF}\left(p^2\right)$ of length n is a right near-subspace $\mathcal{C}\subseteq \mathcal{V}$. Its elements are called codewords. For $\mathbf{v}=(v_1,\dots ,v_n)\in \mathcal{V}$, the Hamming weight of $\mathbf{v}$ is ${\mathrm{wt}}_H\left(\mathbf{v}\right):=\left|\{i\in \{1,\dots ,n\}:v_i\ne 0\}\right|$. The Hamming distance between $\mathbf{x},\mathbf{y}\in \mathcal{V}$ is $d_H(\mathbf{x},\mathbf{y}):={\mathrm{wt}}_H(\mathbf{x}-\mathbf{y})$. If $\mathcal{C}\ne \left\{\mathbf{0}\right\}$, the minimum Hamming distance of $\mathcal{C}$ is $d_H\left(\mathcal{C}\right)=min\{{\mathrm{wt}}_H\left(\mathbf{c}\right):\mathbf{c}\in \mathcal{C},\mathbf{c}\ne \mathbf{0}\}$.

A subset $\{{\mathbf{g}}_1,\dots ,{\mathbf{g}}_k\}\subseteq \mathcal{C}$ is called right independent if ${\sum }_{i=1}^k{\mathbf{g}}_i\circ a_i=\mathbf{0}$ implies $a_1=\dots =a_k=0$. A right basis of $\mathcal{C}$ is a right independent set that generates $\mathcal{C}$ under addition and the coordinate-wise right action ∘. If $\mathcal{C}\ne \left\{\mathbf{0}\right\}$ admits a right basis of size k, we call k the right dimension of $\mathcal{C}$ and write ${dim}_R\left(\mathcal{C}\right)=k$. In this case, we also say that $\mathcal{C}$ is an $[n,k,d_H]$ right-linear code, where $d_H=d_H\left(\mathcal{C}\right)$. We use the Magma [16] notation

$$[<0,1>,···,<i,A_i>,···,<n,A_n>]$$

for the weight distribution of a linear code $\mathcal{C}$ over $\mathrm{NF}\left(p^2\right)$, where $A_i$ is the number of codewords of Hamming weight i.

Lemma 7.

Let $\mathcal{C}\subseteq \mathrm{NF}{\left(p^2\right)}^n$ be a nonzero right-linear code over $\mathrm{NF}\left(p^2\right)$. Then $\mathcal{C}$ admits a right basis. Moreover, if $\{{\mathbf{g}}_1,\dots ,{\mathbf{g}}_k\}$ is a right basis of $\mathcal{C}$, then every $\mathbf{c}\in \mathcal{C}$ can be written uniquely in the form

$$\mathbf{c}=\sum _{i=1}^k{\mathbf{g}}_i\circ a_i(a_1,\dots ,a_k\in \mathrm{NF}\left(p^2\right)),$$

and hence $\left|\mathcal{C}\right|=|\mathrm{NF}\left(p^2\right){|}^k$. In particular, any two right bases of $\mathcal{C}$ have the same cardinality, so ${dim}_R\left(\mathcal{C}\right)$ is well-defined and satisfies ${dim}_R\left(\mathcal{C}\right)\le n$.

Proof. 

Since $\mathrm{NF}\left(p^2\right)$ is finite, $\mathcal{C}$ is finite, hence it is generated by some finite subset under addition and the right action ∘. Choose a generating set $\{{\mathbf{g}}_1,\dots ,{\mathbf{g}}_k\}$ of $\mathcal{C}$ of minimal cardinality. We claim it is right independent. Indeed, if ${\sum }_{i=1}^k{\mathbf{g}}_i\circ a_i=\mathbf{0}$ with some $a_j\ne 0$, then

$${\mathbf{g}}_j\circ a_j=-\sum _{i\ne j}{\mathbf{g}}_i\circ a_i,$$

and multiplying on the right by $a_j^{-1}\in \mathrm{NF}{\left(p^2\right)}^{\times }$ gives

$${\mathbf{g}}_j=-\sum _{i\ne j}{\mathbf{g}}_i\circ (a_i\circ a_j^{-1}),$$

so ${\mathbf{g}}_j$ lies in the right span of $\{{\mathbf{g}}_i:i\ne j\}$, contradicting the minimality of the generating set. Thus $\{{\mathbf{g}}_1,\dots ,{\mathbf{g}}_k\}$ is a right basis.

If ${\sum }_{i=1}^k{\mathbf{g}}_i\circ a_i={\sum }_{i=1}^k{\mathbf{g}}_i\circ b_i$, then by right distributivity ${\sum }_{i=1}^k{\mathbf{g}}_i\circ (a_i-b_i)=\mathbf{0}$, and right independence implies $a_i=b_i$ for all i. Hence, the representation is unique, and the coefficient map $\mathrm{NF}{\left(p^2\right)}^k\to \mathcal{C}$ is a bijection, yielding $\left|\mathcal{C}\right|=|\mathrm{NF}\left(p^2\right){|}^k$. Consequently, k is determined by $\left|\mathcal{C}\right|$, so any two right bases have the same size and ${dim}_R\left(\mathcal{C}\right)$ is well-defined. Finally, $\left|\mathcal{C}\right|\le |\mathrm{NF}\left(p^2\right){|}^n$ implies $k\le n$.   □

3.2. Generator Matrices, Monomial Equivalence, and Systematic Form

Let $\mathcal{C}\subseteq \mathcal{V}$ be a right-linear code over $\mathrm{NF}\left(p^2\right)$ with ${dim}_R\left(\mathcal{C}\right)=k$. A generator matrix of $\mathcal{C}$ is a matrix $G\in \mathrm{NF}{\left(p^2\right)}^{k\times n}$ whose rows form a right basis of $\mathcal{C}$. Equivalently, if ${\mathbf{g}}_1,\dots ,{\mathbf{g}}_k$ are the rows of G, then $\mathcal{C}$ is the right row span of G,

$$\mathcal{C}=\langle G\rangle :=\left\{\sum _{i=1}^k{\mathbf{g}}_i\circ a_i:a_1,\dots ,a_k\in \mathrm{NF}\left(p^2\right)\right\}.$$

Thus every right-linear code over $\mathrm{NF}\left(p^2\right)$ of length n and right dimension k can be realized as the right row span of a $k\times n$ matrix over $\mathrm{NF}\left(p^2\right)$.

An $n\times n$ matrix N over $\mathrm{NF}\left(p^2\right)$ is called a monomial matrix if each row and each column contains exactly one nonzero entry, and every nonzero entry lies in ${\mathbb{F}}_p^{\times }$. For a row vector $\mathbf{c}=(c_1,\dots ,c_n)\in \mathrm{NF}{\left(p^2\right)}^n$ we define the right action of N by

$${\left(\mathbf{c}N\right)}_j=\sum _{i=1}^n{c}_i\circ N_{ij}(j=1,\dots ,n),$$

where products are computed using the Dickson product ∘. Two right-linear codes $\mathcal{C},{\mathcal{C}}^{\prime }\subseteq \mathrm{NF}{\left(p^2\right)}^n$ are right monomially equivalent if there exists a monomial matrix N such that

$${\mathcal{C}}^{\prime }=\mathcal{C}N:=\{\mathbf{c}N:\mathbf{c}\in \mathcal{C}\}.$$

A basic step in the structural study of right-linear codes over a right near-field is to bring generator matrices into a standard (systematic-type) form, analogous to the classical systematic form over finite fields (see, e.g., [2]). Since left distributivity is not available, in general, the reduction uses only right row operations together with column permutations induced by right monomial transformations. This separates an information set from the parity part and streamlines comparisons up to right monomial equivalence.

Theorem 1.

Every right-linear code $\mathcal{C}$ with parameters $[n,k,d_H]$ is right monomially equivalent to a code admitting a systematic generator matrix of the form $G=\left[I_{k}A\right]$, for some $A\in \mathrm{NF}{\left(p^2\right)}^{k\times (n-k)}$.

Proof. 

Let $G_0\in \mathrm{NF}{\left(p^2\right)}^{k\times n}$ be a generator matrix of $\mathcal{C}$ (with $k={dim}_R\left(\mathcal{C}\right)$). For a nonzero $x\in \mathrm{NF}\left(p^2\right)$ we write $x^{-1}$ for the inverse of x with respect to ∘; thus $x\circ x^{-1}=1=x^{-1}\circ x$.

The following right elementary row operations preserve the right row span

$$\langle G_0\rangle =\left\{\sum _{i=1}^k{R}_i\circ b_i:b_i\in \mathrm{NF}\left(p^2\right)\right\},$$

where $R_i$ denotes the ith row of $G_0$:

(i)

swap two rows;

(ii)

right-scale a row by a unit $u\in \mathrm{NF}{\left(p^2\right)}^{\times }$, i.e., $R_i\leftarrow R_i\circ u$;

(iii)

replace $R_i$ by $R_i+R_j\circ a$ for some $a\in \mathrm{NF}\left(p^2\right)$.

Indeed, (i) is immediate. In (ii), the map $b\mapsto b\circ u$ is bijective on $\mathrm{NF}\left(p^2\right)$ since u is invertible under ∘, so right-scaling does not change the set of right linear combinations. In (iii), right distributivity of ∘ in each coordinate shows that right linear combinations of the new rows are exactly the right linear combinations of the old rows.

We now perform Gaussian elimination using only (i)–(iii). In any nonzero column, choose a nonzero entry p and swap its row into the current pivot position. Since $p\ne 0$, there exists $v=p^{-1}\in \mathrm{NF}{\left(p^2\right)}^{\times }$ with $p\circ v=1$; right-scale the pivot row by v so that the pivot becomes 1. For any other row having entry e in the pivot column, replace that row by

$$R_{\mathrm{new}}=R_{\mathrm{old}}+R_{\mathrm{pivot}}\circ (-e).$$

In the pivot column the new entry equals

$$e+\left(1\circ (-e)\right)=e+{\sigma }_{-e}\left(1\right)(-e)=e+(-e)=0,$$

since ${\sigma }_y\left(1\right)=1$ for all $y\in \mathrm{NF}\left(p^2\right)$. Iterating down the columns yields a row-echelon form with k pivot columns and pivots equal to 1.

Finally, permute columns so that the pivot columns come first, obtaining a systematic generator matrix

$$G=\left[I_{k}A\right].$$

This column permutation is induced by a permutation matrix, which is a monomial matrix (its nonzero entries are $1\in {\mathbb{F}}_p^{\times }$), hence the resulting code is right monomially equivalent to $\mathcal{C}$.   □

Example 4.

Let $\mathrm{NF}\left(p^2\right)=\mathrm{NF}\left(25\right)$ and consider the right-linear code $\mathcal{C}\subseteq \mathrm{NF}{\left(p^2\right)}^4$ generated by

$$G_0=\left[\begin{array}{cccc}1& 2& 0& 1\\ 0& 1& 1& 2\end{array}\right]\in \mathrm{NF}{\left(p^2\right)}^{2\times 4}.$$

Take the first two columns

$$P=\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right],$$

which is invertible. Its inverse is

$$P^{-1}=\left[\begin{array}{cc}1& 3\\ 0& 1\end{array}\right]$$

Left-multiplying by $P^{-1}$ gives an equivalent generator matrix

$$G=P^{-1}{G}_0=\left[\begin{array}{cc}1& 3\\ 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 2& 0& 1\\ 0& 1& 1& 2\end{array}\right]=\left[\begin{array}{cccc}1& 0& 3& 2\\ 0& 1& 1& 2\end{array}\right].$$

Hence G is systematic:

$$G=\left[I_{2}A\right],A=\left[\begin{array}{cc}3& 2\\ 1& 2\end{array}\right]\in \mathrm{NF}{\left(p^2\right)}^{2\times 2}.$$

Remark 3.

Although the displayed generator matrix G has entries in ${\mathbb{F}}_p$, the code $\mathcal{C}$ is a right $\mathrm{NF}\left(p^2\right)$-linear code, so it is generated by right linear combinations with coefficients in $\mathrm{NF}\left(p^2\right)$. In particular, let $\alpha \in \mathrm{NF}\left(p^2\right)\setminus {\mathbb{F}}_p$ and let $r_i$ be any nonzero row of G. Then $r_i\circ \alpha \in \mathcal{C}$. Moreover, since $r_i$ has at least one nonzero entry, say $g_{ij}\in {\mathbb{F}}_p^{\times }$, we obtain a coordinate outside the prime field:

$${(r_i\circ \alpha )}_j=g_{ij}\circ \alpha ={\sigma }_{\alpha }\left(g_{ij}\right)\alpha =g_{ij}\alpha \notin {\mathbb{F}}_p,$$

because ${\sigma }_{\alpha }\in \{\mathrm{id},\phi \}$ fixes ${\mathbb{F}}_p$ and $\alpha \notin {\mathbb{F}}_p$.

For example, take $\omega \in mathrmNF\left(25\right)\setminus {\mathbb{F}}_5$. From $r_1=(1,0,3,2)$ we get

$$r_1\circ \omega =(\omega ,0,3\omega ,2\omega )\in \mathcal{C}\setminus {\mathbb{F}}_5^4.$$

In particular, $\mathrm{NF}\left(p^2\right)\left(25\right)$ is not a field: left distributivity fails. For example, let $u=1$ and $v=2\omega$. Then $v\notin H_0$ and $u+v=1+2\omega \in H_0$ since ${(2+3\omega )}^2=1+2\omega$. Also ${\omega }^5=4\omega$, so

$$\omega \circ (u+v)=\omega \circ (1+2\omega )=\omega (1+2\omega )=\omega +1,$$

whereas

$$\omega \circ u+\omega \circ v=\omega \circ 1+\omega \circ \left(2\omega \right)=\omega +{\omega }^5\left(2\omega \right)=\omega +4.$$

Thus $\omega \circ (u+v)\ne \omega \circ u+\omega \circ v$, and $\mathrm{N}F\left(25\right)$ is not a field. We also refer to Example 2 in [12] for a complementary illustration in $\mathrm{N}F\left(9\right)$, where the generator matrix itself has entries from $\mathrm{N}F\left(9\right)\setminus {\mathbb{F}}_3$.

Lemma 8.

Let $G=\left[I_{k}A\right]\in \mathrm{NF}{\left(p^2\right)}^{k\times n}$ and $\mathcal{C}=\langle G\rangle$. Let $\pi :\mathrm{NF}{\left(p^2\right)}^n\to \mathrm{NF}{\left(p^2\right)}^k$ be the projection onto the first k coordinates. Then ${\pi |}_{\mathcal{C}}:\mathcal{C}\to \mathrm{NF}{\left(p^2\right)}^k$ is a bijection. More precisely, for every $b\in \mathrm{NF}{\left(p^2\right)}^k$ there is a unique codeword $\mathbf{c}\in \mathcal{C}$ whose first k coordinates equal b, namely

$$\mathrm{enc}\left(b\right):=(b,Ab)\in \mathcal{C},{\left(Ab\right)}_j:=\sum _{i=1}^k{A}_{ij}\circ b_i(j=1,\dots ,n-k).$$

In particular, for each $i\in \{1,\dots ,k\}$ there is a unique codeword in $\mathcal{C}$ whose first k coordinates equal $e_i$.

Proof. 

Write the rows of G as ${\mathbf{r}}_i=(e_i,{\mathbf{a}}_i)$, where ${\mathbf{a}}_i$ is the ith row of A. For $b=(b_1,\dots ,b_k)\in \mathrm{NF}{\left(p^2\right)}^k$, consider

$$\mathbf{c}=\sum _{i=1}^k{\mathbf{r}}_i\circ b_i\in \mathcal{C}.$$

Using the coordinate-wise right action, the first k coordinates of $\mathbf{c}$ are ${\sum }_{i=1}^k{e}_i\circ b_i=b$, and the last $n-k$ coordinates are ${\sum }_{i=1}^k{\mathbf{a}}_i\circ b_i=Ab$. Hence, $\mathbf{c}=(b,Ab)=\mathrm{enc}\left(b\right)$, so $\pi \left(\mathrm{enc}\right(b\left)\right)=b$ and ${\pi |}_{\mathcal{C}}$ is surjective. If $\mathrm{enc}\left(b\right)=(0,0)$ then its first k coordinates give $b=0$, so $\mathrm{enc}$ is injective. Therefore ${\pi |}_{\mathcal{C}}$ is bijective with inverse $\mathrm{enc}$, and the uniqueness statement follows by taking $b=e_i$.   □

Corollary 3.

Let $\mathcal{C}$ admit a systematic generator matrix with information set $\{1,\dots ,k\}$. If $\left[I_{k}A\right]$ and $\left[I_k{A}^{\prime }\right]$ both generate $\mathcal{C}$, then $A=A^{\prime }$.

Proof. 

By Lemma 8, for each i the unique codeword in $\mathcal{C}$ whose first k coordinates equal $e_i$ is the ith row of any systematic generator matrix with this information set. Hence, the ith rows of A and $A^{\prime }$ agree for all i.   □

Corollary 4.

Let $\mathcal{C}$ be a right-linear code with parameters $[n,k,d_H]$. Then $\left|\mathcal{C}\right|=p^{2k}$.

Proof. 

Let $G\in \mathrm{NF}{\left(p^2\right)}^{k\times n}$ be a generator matrix of $\mathcal{C}$ with rows ${\mathbf{g}}_1,\dots ,{\mathbf{g}}_k$. Define $\varphi :\mathrm{NF}{\left(p^2\right)}^k\to \mathcal{C}$ by $\varphi (u_1,\dots ,u_k)={\sum }_{i=1}^k{\mathbf{g}}_i\circ u_i$. By definition of generator matrix, $\varphi$ is surjective. If $\varphi (u_1,\dots ,u_k)=\mathbf{0}$, then ${\sum }_{i=1}^k{\mathbf{g}}_i\circ u_i=\mathbf{0}$, and right independence of ${\mathbf{g}}_1,\dots ,{\mathbf{g}}_k$ forces $u_1=\dots =u_k=0$. Hence, $\varphi$ is injective, thus bijective, and $\left|\mathcal{C}\right|=|\mathrm{NF}{\left(p^2\right)}^k|=|\mathrm{NF}\left(p^2\right){|}^k=p^{2k}$.   □

3.3. Properties of Galois Conjugation

We summarize some basic properties of Galois conjugation on $\mathrm{NF}{\left(p^2\right)}^n$.

Corollary 5.

Let $\mathcal{C}\subseteq \mathrm{NF}{\left(p^2\right)}^n$ be a right-linear code over $\mathrm{NF}\left(p^2\right)$. Then:

 (i) 

$\overline{\mathcal{C}}$ is a right-linear code over $\mathrm{NF}\left(p^2\right)$;

 (ii) 

$|\overline{\mathcal{C}}|=|\mathcal{C}|$;

 (iii) 

${dim}_R\left(\overline{\mathcal{C}}\right)={dim}_R\left(\mathcal{C}\right)$.

Proof. 

(i)

Conjugation $\overline{\phantom{x}}:\mathrm{NF}{\left(p^2\right)}^n\to \mathrm{NF}{\left(p^2\right)}^n$ is an ${\mathbb{F}}_p$–linear bijection, hence an automorphism of the additive group $(\mathrm{NF}{\left(p^2\right)}^n,+)$. Therefore $\overline{\mathcal{C}}$ is an additive subgroup of $\mathrm{NF}{\left(p^2\right)}^n$. Let $\mathbf{u}\in \overline{\mathcal{C}}$ and $a\in \mathrm{NF}\left(p^2\right)$, and write $\mathbf{u}=\overline{\mathbf{c}}$ with $\mathbf{c}\in \mathcal{C}$. Using Lemma 4(e) with a replaced by $\overline{a}$ and $\overline{\overline{a}}=a$, we get

$$\mathbf{u}\circ a=\overline{\mathbf{c}}\circ a=\overline{\mathbf{c}\circ \overline{a}}\in \overline{\mathcal{C}},$$

since $\mathbf{c}\circ \overline{a}\in \mathcal{C}$ by right-linearity of $\mathcal{C}$.

(ii)

Conjugation is a bijection on $\mathrm{NF}{\left(p^2\right)}^n$, hence restricts to a bijection $\mathcal{C}\to \overline{\mathcal{C}}$. Thus $|\overline{\mathcal{C}}|=|\mathcal{C}|$.

(iii)

Let $\{{\mathbf{g}}_1,\dots ,{\mathbf{g}}_k\}$ be a right basis of $\mathcal{C}$. Then $\{\overline{{\mathbf{g}}_1},\dots ,\overline{{\mathbf{g}}_k}\}$ is a right basis of $\overline{\mathcal{C}}$ (using Lemma 4(e)), hence ${dim}_R\left(\overline{\mathcal{C}}\right)={dim}_R\left(\mathcal{C}\right)$.

□

Lemma 9.

For all $\mathbf{x},\mathbf{y}\in \mathrm{NF}{\left(p^2\right)}^n$, one has ${\mathrm{wt}}_H\left(\overline{\mathbf{x}}\right)={\mathrm{wt}}_H\left(\mathbf{x}\right)$ and $d_H(\overline{\mathbf{x}},\overline{\mathbf{y}})=d_H(\mathbf{x},\mathbf{y})$. In particular, Galois conjugation is a Hamming isometry of $\mathrm{NF}{\left(p^2\right)}^n$.

Proof. 

Since $\overline{\phantom{x}}:\mathrm{NF}\left(p^2\right)\to \mathrm{NF}\left(p^2\right)$ is a bijection with $\overline{0}=0$, we have $\overline{x_i}=0$ if and only if $x_i=0$ for each i, and hence ${\mathrm{wt}}_H\left(\overline{\mathbf{x}}\right)={\mathrm{wt}}_H\left(\mathbf{x}\right)$. Moreover, conjugation is additive, so $\overline{\mathbf{x}}-\overline{\mathbf{y}}=\overline{\mathbf{x}-\mathbf{y}}$ coordinate-wise. Therefore

$$d_H(\overline{\mathbf{x}},\overline{\mathbf{y}})={\mathrm{wt}}_H(\overline{\mathbf{x}}-\overline{\mathbf{y}})={\mathrm{wt}}_H\left(\overline{\mathbf{x}-\mathbf{y}}\right)={\mathrm{wt}}_H(\mathbf{x}-\mathbf{y})=d_H(\mathbf{x},\mathbf{y}).$$

□

Proposition 3.

Let $\mathcal{C}\subseteq \mathrm{NF}{\left(p^2\right)}^n$ be a right-linear code with generator matrix $G\in \mathrm{NF}{\left(p^2\right)}^{k\times n}$. Then the following are equivalent:

 (i) 

$\mathcal{C}=\overline{\mathcal{C}}$.

 (ii) 

$\langle G\rangle =\langle \overline{G}\rangle$.

Proof. 

Since $\mathcal{C}=\langle G\rangle$, Lemma 5 gives $\overline{\mathcal{C}}=\overline{\langle G\rangle }=\langle \overline{G}\rangle$. Thus $\mathcal{C}=\overline{\mathcal{C}}$ if and only if $\langle G\rangle =\langle \overline{G}\rangle$.   □

Remark 4.

By Proposition 3, the entry-wise condition $G=\overline{G}$ is sufficient but not necessary for $\mathcal{C}=\overline{\mathcal{C}}$. Since the fixed field of $\overline{x}=x^p$ on ${\mathbb{F}}_{p^2}$ is ${\mathbb{F}}_p$, the equality $G=\overline{G}$ holds, for instance, whenever all entries of G lie in ${\mathbb{F}}_p$.

The converse need not hold: a right-linear code may satisfy $\mathcal{C}=\overline{\mathcal{C}}$ even though a chosen generator matrix is not fixed entry-wise. For example, for $n=2$ consider $\mathcal{C}=\{(x,0)\in \mathrm{NF}{\left(p^2\right)}^2:x\in \mathrm{NF}\left(p^2\right)\}$. Then $\overline{\mathcal{C}}=\mathcal{C}$. Choosing $a\in \mathrm{NF}\left(p^2\right)\setminus {\mathbb{F}}_p$, the matrix $G=\left[a0\right]$ generates $\mathcal{C}$ but satisfies $G\ne \overline{G}$.

Corollary 6.

If $G=\left[I_{k}A\right]$ is a systematic generator matrix of $\mathcal{C}$, then $\overline{G}=\left[I_k\overline{A}\right]$ is a systematic generator matrix of $\overline{\mathcal{C}}$.

Proof. 

This follows from $\overline{0}=0$, $\overline{1}=1$, and Lemma 5.   □

3.4. Right Monomial Matrices

Right monomial matrices represent coordinate permutations and ${\mathbb{F}}_p^{\times }$-reweightings compatible with the Dickson right action. They preserve the Hamming metric and behave well with conjugation and left-stability. We record these facts and then describe their effect on the parity block A in a systematic generator matrix $G=\left[I_{k}A\right]$.

Lemma 10.

Let $N\in \mathrm{NF}{\left(p^2\right)}^{n\times n}$ be a monomial matrix. Then $N=PD$, where P is a permutation matrix and $D=\mathrm{diag}(c_1,\dots ,c_n)$ with $c_j\in {\mathbb{F}}_p^{\times }$. In particular, monomial matrices form a group under matrix multiplication, and $N^{-1}$ is monomial.

Proof. 

Each column j has a unique nonzero entry $N_{\pi \left(j\right),j}=c_j\in {\mathbb{F}}_p^{\times }$; defining $P_{\pi \left(j\right),j}=1$ and $D=\mathrm{diag}(c_1,\dots ,c_n)$ gives $N=PD$. The remaining claims follow immediately.   □

Lemma 11.

Let $N\in \mathrm{NF}{\left(p^2\right)}^{n\times n}$ be a monomial matrix. Then there exist a permutation π of $\{1,\dots ,n\}$ and scalars $c_1,\dots ,c_n\in {\mathbb{F}}_p^{\times }$ such that for every $\mathbf{v}=(v_1,\dots ,v_n)\in \mathrm{NF}{\left(p^2\right)}^n$,

$${\left(\mathbf{v}N\right)}_j=v_{\pi \left(j\right)}\circ c_j(j=1,\dots ,n).$$

Proof. 

Immediate from the definition: in column j the only nonzero entry is $N_{\pi \left(j\right),j}=c_j\in {\mathbb{F}}_p^{\times }$, hence

$${\left(\mathbf{v}N\right)}_j=\sum _{i=1}^n{v}_i\circ N_{ij}=v_{\pi \left(j\right)}\circ c_j.$$

□

Lemma 12.

For all $a\in \mathrm{NF}\left(p^2\right)$ and all $c\in {\mathbb{F}}_p^{\times }$, one has $c\circ a=a\circ c$. Consequently, for all $x\in \mathrm{NF}\left(p^2\right)$, $(x\circ c)\circ a=(x\circ a)\circ c$.

Proof. 

If $a=0$ the claim is trivial. Assume $a\ne 0$. Since $c\in {\mathbb{F}}_p^{\times }$, we have ${\sigma }_a\left(c\right)=c$ and hence $c\circ a={\sigma }_a\left(c\right)a=ca$. Also $c\in {\mathbb{F}}_p^{\times }\subseteq H_0$, so ${\sigma }_c=\mathrm{id}$ and $a\circ c={\sigma }_c\left(a\right)c=ac=ca$. Associativity of ∘ yields $(x\circ c)\circ a=x\circ (c\circ a)=x\circ (a\circ c)=(x\circ a)\circ c$.   □

Lemma 13.

Let N be a right monomial matrix. Then for all $\mathbf{v}\in \mathrm{NF}{\left(p^2\right)}^n$ and all $a\in \mathrm{NF}\left(p^2\right)$, $\left(\mathbf{v}N\right)\circ a=(\mathbf{v}\circ a)N$.

Proof. 

By the observation above, there exist a permutation $\pi$ of $\{1,\dots ,n\}$ and scalars $c_1,\dots ,c_n\in {\mathbb{F}}_p^{\times }$ such that ${\left(\mathbf{v}N\right)}_j=v_{\pi \left(j\right)}\circ c_j$ for $j=1,\dots ,n$.

Fix j. Using associativity of ∘ and Lemma 12, we compute

$$\begin{array}{cc}\hfill {\left(\left(\mathbf{v}N\right)\circ a\right)}_j& ={\left(\mathbf{v}N\right)}_j\circ a=(v_{\pi \left(j\right)}\circ c_j)\circ a=(v_{\pi \left(j\right)}\circ a)\circ c_j.\hfill \end{array}$$

On the other hand,

$$\begin{array}{cc}\hfill {\left((\mathbf{v}\circ a)N\right)}_j& =\sum _{i=1}^n(v_i\circ a)\circ N_{ij}=(v_{\pi \left(j\right)}\circ a)\circ c_j,\hfill \end{array}$$

since the jth column of N has the unique nonzero entry $N_{\pi \left(j\right),j}=c_j$. Thus ${\left(\left(\mathbf{v}N\right)\circ a\right)}_j={\left((\mathbf{v}\circ a)N\right)}_j$ for every j, and hence $\left(\mathbf{v}N\right)\circ a=(\mathbf{v}\circ a)N$.   □

Corollary 7.

Let $\mathcal{C}\subseteq \mathrm{NF}{\left(p^2\right)}^n$ be a right near-subspace and let N be a right monomial matrix. Then $\mathcal{C}N$ is a right near-subspace. Moreover, if $\mathcal{C}=\langle G\rangle$ for some $G\in \mathrm{NF}{\left(p^2\right)}^{k\times n}$, then $\mathcal{C}N=\langle GN\rangle$.

Proof. 

First, $\mathcal{C}N$ is an additive subgroup: if $\mathbf{v},\mathbf{w}\in \mathcal{C}$, then for each j,

$$\begin{array}{cc}\hfill {\left((\mathbf{v}+\mathbf{w})N\right)}_j& =\sum _{i=1}^n(v_i+w_i)\circ N_{ij}=\sum _{i=1}^n{v}_i\circ N_{ij}+\sum _{i=1}^n{w}_i\circ N_{ij}={\left(\mathbf{v}N\right)}_j+{\left(\mathbf{w}N\right)}_j,\hfill \end{array}$$

so $(\mathbf{v}+\mathbf{w})N=\mathbf{v}N+\mathbf{w}N\in \mathcal{C}N$, and similarly $(-\mathbf{v})N=-\left(\mathbf{v}N\right)$. Next, for right-closure let $\mathbf{v}N\in \mathcal{C}N$ with $\mathbf{v}\in \mathcal{C}$ and $a\in \mathrm{NF}\left(p^2\right)$. By Lemma 13,

$$\left(\mathbf{v}N\right)\circ a=(\mathbf{v}\circ a)N\in \mathcal{C}N,$$

since $\mathbf{v}\circ a\in \mathcal{C}$. Hence, $\mathcal{C}N$ is a right near-subspace.

If $\mathcal{C}=\langle G\rangle$ and G has rows ${\mathbf{g}}_1,\dots ,{\mathbf{g}}_k$, then any element of $\mathcal{C}N$ is

$$\begin{array}{cc}\hfill \left(\sum _{i=1}^k{\mathbf{g}}_i\circ a_i\right)N& =\sum _{i=1}^k({\mathbf{g}}_i\circ a_i)N=\sum _{i=1}^k\left({\mathbf{g}}_{i}N\right)\circ a_i\in \langle GN\rangle ,\hfill \end{array}$$

using Lemma 13 in the last step; the reverse inclusion is immediate.   □

Proposition 4.

Let N be a right monomial matrix. Then for all $\mathbf{x},\mathbf{y}\in \mathrm{NF}{\left(p^2\right)}^n$,

$${\mathrm{wt}}_H\left(\mathbf{x}N\right)={\mathrm{wt}}_H\left(\mathbf{x}\right),\mathit{and}{d}_H(\mathbf{x}N,\mathbf{y}N)=d_H(\mathbf{x},\mathbf{y}).$$

Consequently, if ${\mathcal{C}}^{\prime }=\mathcal{C}N$, then $d_H\left({\mathcal{C}}^{\prime }\right)=d_H\left(\mathcal{C}\right)$.

Proof. 

Write ${\left(\mathbf{x}N\right)}_j=x_{\pi \left(j\right)}\circ c_j$ as in Lemma 11. Since $c_j\ne 0$, right multiplication by $c_j$ is a bijection on $\mathrm{NF}\left(p^2\right)$, hence $x_{\pi \left(j\right)}\circ c_j=0$ if and only if $x_{\pi \left(j\right)}=0$. Thus the support is permuted by $\pi$, and ${\mathrm{wt}}_H\left(\mathbf{x}N\right)={\mathrm{wt}}_H\left(\mathbf{x}\right)$. The distance statement follows from $\mathbf{x}N-\mathbf{y}N=(\mathbf{x}-\mathbf{y})N$ and the weight invariance.   □

Corollary 8.

Let $\mathcal{C}\subseteq \mathrm{NF}{\left(p^2\right)}^n$ be a right-linear code and let N be a right monomial matrix. Set ${\mathcal{C}}^{\prime }:=\mathcal{C}N$. Then:

 (i) 

${dim}_R\left({\mathcal{C}}^{\prime }\right)={dim}_R\left(\mathcal{C}\right)$;

 (ii) 

$|{\mathcal{C}}^{\prime }|=|\mathcal{C}|$;

 (iii) 

$d_H\left({\mathcal{C}}^{\prime }\right)=d_H\left(\mathcal{C}\right)$.

Proof. 

Let $T_N:\mathcal{V}\to \mathcal{V}$ be the map $T_N\left(\mathbf{x}\right)=\mathbf{x}N$. Since N is invertible (Lemma 10), $T_N$ is a bijection. Hence, $|{\mathcal{C}}^{\prime }|=|\mathcal{C}|$, proving (ii).

By Corollary 7, ${\mathcal{C}}^{\prime }$ is a right near-subspace and $T_N$ restricts to a bijection $\mathcal{C}\to {\mathcal{C}}^{\prime }$. Therefore the image under $T_N$ of any right basis of $\mathcal{C}$ is a right basis of ${\mathcal{C}}^{\prime }$, so ${dim}_R\left({\mathcal{C}}^{\prime }\right)={dim}_R\left(\mathcal{C}\right)$, proving (i).

Finally, Proposition 4 shows that $T_N$ preserves Hamming distance, and thus $d_H\left({\mathcal{C}}^{\prime }\right)=d_H\left(\mathcal{C}\right)$, proving (iii).   □

Lemma 14.

Let N be a right monomial matrix. Then $\overline{N}=N$ and, for all $\mathbf{x}\in \mathrm{NF}{\left(p^2\right)}^n$,

$$\overline{\mathbf{x}N}=\overline{\mathbf{x}}N.$$

Consequently, for any $\mathcal{C}\subseteq \mathrm{NF}{\left(p^2\right)}^n$ one has $\overline{\mathcal{C}N}=\overline{\mathcal{C}}N$.

Proof. 

All nonzero entries of N lie in ${\mathbb{F}}_p^{\times }$ and are fixed by conjugation, hence $\overline{N}=N$. Using Lemma 11 and Lemma 4(d), for each j,

$$\begin{array}{cc}\hfill \overline{{\left(\mathbf{x}N\right)}_j}& =\overline{x_{\pi \left(j\right)}\circ c_j}=\overline{x_{\pi \left(j\right)}}\circ c_j={\left(\left(\overline{\mathbf{x}}\right)N\right)}_j,\hfill \end{array}$$

so $\overline{\mathbf{x}N}=\overline{\mathbf{x}}N$. The set identity follows immediately.   □

Corollary 9.

If $\mathcal{C}=\overline{\mathcal{C}}$ and N is a right monomial matrix, then $\mathcal{C}N=\overline{\mathcal{C}N}$.

Proof. 

By Lemma 14, $\overline{\mathcal{C}N}=\overline{\mathcal{C}}N=\mathcal{C}N$.   □

Lemma 15.

Let $\mathcal{C}\subseteq \mathrm{NF}{\left(p^2\right)}^n$ be left-stable and let N be a right monomial matrix. Then $\mathcal{C}N$ is left-stable.

Proof. 

Write ${\left(\mathbf{v}N\right)}_j=v_{\pi \left(j\right)}\circ c_j$ as in Lemma 11. Let $a\in \mathrm{NF}\left(p^2\right)$ and $\mathbf{v}\in \mathcal{C}$. By associativity of ∘,

$$\begin{array}{cc}\hfill {\left(a\circ \left(\mathbf{v}N\right)\right)}_j& =a\circ \left(v_{\pi \left(j\right)}\circ c_j\right)=(a\circ v_{\pi \left(j\right)})\circ c_j={\left((a\circ \mathbf{v})N\right)}_j.\hfill \end{array}$$

Hence $a\circ \left(\mathbf{v}N\right)=(a\circ \mathbf{v})N\in \mathcal{C}N$ since $a\circ \mathbf{v}\in \mathcal{C}$.   □

Theorem 2.

Let $G=\left[I_{k}A\right]$ and $G^{\prime }=\left[I_k{A}^{\prime }\right]$ be systematic generator matrices in $\mathrm{NF}{\left(p^2\right)}^{k\times n}$ with the same information set $\{1,\dots ,k\}$, and set $\mathcal{C}=\langle G\rangle$ and ${\mathcal{C}}^{\prime }=\langle G^{\prime }\rangle$. Then the following are equivalent:

 (i) 

${\mathcal{C}}^{\prime }=\mathcal{C}N$ for some right monomial matrix N that fixes the first k coordinates, i.e., ${\left(\mathbf{x}N\right)}_j=x_j$ for all $\mathbf{x}\in \mathrm{NF}{\left(p^2\right)}^n$ and all $j=1,\dots ,k$.

 (ii) 

There exists a right monomial matrix $M\in \mathrm{NF}{\left(p^2\right)}^{(n-k)\times (n-k)}$ such that $A^{\prime }=AM$, where

$${\left(AM\right)}_{ij}:=\sum _{\ell =1}^{n-k}{A}_{i\ell }\circ M_{\ell j}.$$

Equivalently,

$$G^{\prime }=G\left[\begin{array}{cc}{I}_k& 0\\ 0& M\end{array}\right].$$

Proof. 

(ii)⇒(i). Assume $A^{\prime }=AM$ for a right monomial matrix M, and set $N=\left[\begin{array}{cc}{I}_k& 0\\ 0& M\end{array}\right]$. Then N is right monomial and fixes the first k coordinates. Moreover,

$$\begin{array}{cc}\hfill GN& =\left[I_{k}A\right]\left[\begin{array}{cc}{I}_k& 0\\ 0& M\end{array}\right]=\left[I_{k}AM\right]=\left[I_k{A}^{\prime }\right]=G^{\prime },\hfill \end{array}$$

so ${\mathcal{C}}^{\prime }=\langle G^{\prime }\rangle =\langle GN\rangle =\langle G\rangle N=\mathcal{C}N$.

(i)⇒(ii). Assume ${\mathcal{C}}^{\prime }=\mathcal{C}N$ for a right monomial matrix N fixing the first k coordinates. The fixing condition implies that the first k columns of N are the standard basis columns, hence N has block form $N=\left[\begin{array}{cc}{I}_k& 0\\ 0& M\end{array}\right]$ with M right monomial. Therefore $GN=\left[I_{k}AM\right]$ and ${\mathcal{C}}^{\prime }=\langle GN\rangle$. Since both $GN$ and $G^{\prime }$ are systematic with the same information set and generate the same code ${\mathcal{C}}^{\prime }$, Corollary 3 forces $A^{\prime }=AM$.   □

3.5. A Basic Bound and a Left-Stability Criterion

As in the classical setting over finite fields, right-linear codes over $\mathrm{NF}\left(p^2\right)$ satisfy the Singleton bound. The only additional ingredient in our context is the size formula $\left|\mathcal{C}\right|=p^{2k}$ from Corollary 4.

Theorem 3.

Let $\mathcal{C}\subseteq \mathrm{NF}{\left(p^2\right)}^n$ be a right-linear code of right dimension k. Then

$$d_H\left(\mathcal{C}\right)\le n-k+1.$$

Proof. 

Let $d=d_H\left(\mathcal{C}\right)$. Puncturing $\mathcal{C}$ in any set of $d-1$ coordinates is injective: if two codewords agree on the remaining $n-d+1$ coordinates, then their difference is supported on at most $d-1$ positions and must be 0 by the definition of d. Therefore,

$$\left|\mathcal{C}\right|\le |\mathrm{NF}\left(p^2\right){|}^{n-d+1}=p^{2(n-d+1)}.$$

Using $\left|\mathcal{C}\right|=p^{2k}$ from Corollary 4 yields $p^{2k}\le p^{2(n-d+1)}$, hence $k\le n-d+1$ and thus $d\le n-k+1$.   □

A right-linear code $\mathcal{C}\subseteq \mathrm{NF}{\left(p^2\right)}^n$ of right dimension k is called maximum distance separable (MDS) if it attains the Singleton bound, i.e.,

$$d_H\left(\mathcal{C}\right)=n-k+1.$$

Example 5.

From Example 4, the Singleton bound gives $d_H\left(\mathcal{C}\right)\le 4-2+1=3$. We now show that equality holds.

A generic codeword is a right $\mathrm{NF}\left(p^2\right)$–linear combination of the rows of G, hence for $(a,b)\in \mathrm{NF}{\left(p^2\right)}^2$ we have

$$\mathbf{c}(a,b)=(a,b)G=\left(a,b,3\circ a+b,2\circ a+2\circ b\right).$$

Assume that $\mathbf{c}(a,b)$ has Hamming weight $\le 2$. Then at least two coordinates are 0. If $a=0$, then the fourth coordinate becomes $2\circ b$, so $2\circ b=0$ forces $b=0$ (since $\mathrm{NF}\left(p^2\right)$ is a near-field and $2\ne 0$). Similarly, if $b=0$, then the third coordinate is $3\circ a$, so $3\circ a=0$ forces $a=0$. If instead the last two coordinates are 0, then $3\circ a+b=0$ and $2\circ a+2\circ b=0$ imply $b=-3\circ a$ and $a=-b$, hence $a=3\circ a$, so $(3-1)\circ a=2\circ a=0$, which again forces $a=0$ and then $b=0$. Thus no nonzero codeword can have weight 1 or 2, and therefore $d_H\left(\mathcal{C}\right)\ge 3$.

On the other hand, taking $(a,b)=(1,0)$ gives $\mathbf{c}(1,0)=(1,0,3,2)$, which has weight 3. Hence, $d_H\left(\mathcal{C}\right)=3$, and $\mathcal{C}$ meets the Singleton bound with equality.

Theorem 4.

Let $\mathcal{C}$ be a right-linear code with a systematic generator matrix $G=\left[I_{k}A\right]$, where $A\in {\mathbb{F}}_p^{k\times (n-k)}$. Assume that each column of A has at most one nonzero entry. Then $\mathcal{C}$ is left-stable.

Proof. 

Let $a\in \mathrm{NF}\left(p^2\right)$ and $b=(b_1,\dots ,b_k)\in \mathrm{NF}{\left(p^2\right)}^k$. Since $\mathcal{C}=\left\{\right(b,Ab\left)\right\}$, we have

$$a\circ (b,Ab)=\left(a\circ b,a\circ \left(Ab\right)\right).$$

Thus it suffices to prove $a\circ \left(Ab\right)=A(a\circ b)$, where

$${\left(Ab\right)}_j=\sum _{i=1}^k{A}_{ij}\circ b_i,{\left(A(a\circ b)\right)}_j=\sum _{i=1}^k{A}_{ij}\circ (a\circ b_i).$$

Fix $j\in \{1,\dots ,n-k\}$. If the jth column of A is zero then both sides are 0. Otherwise there is a unique index $i_j$ with $A_{i_{j}j}\ne 0$, so ${\left(Ab\right)}_j=A_{i_{j}j}\circ b_{i_j}$. Since $A_{i_{j}j}\in {\mathbb{F}}_p^{\times }\subseteq H_0$, Lemma 3 (b) (with $c=A_{i_{j}j}$) gives

$$a\circ {\left(Ab\right)}_j=a\circ \left(A_{i_{j}j}\circ b_{i_j}\right)=A_{i_{j}j}\circ \left(a\circ b_{i_j}\right)={\left(A(a\circ b)\right)}_j.$$

Hence $a\circ \left(Ab\right)=A(a\circ b)$ coordinate-wise, and therefore $a\circ (b,Ab)=\left(a\circ b,A(a\circ b)\right)\in \mathcal{C}$.   □

Example 6.

Let ${\mathcal{C}}_{\mathrm{rep}}=\{(t,\dots ,t)\in \mathrm{NF}{\left(p^2\right)}^n:t\in \mathrm{NF}\left(p^2\right)\},$ with generator matrix

$$G=\left[1\underset{n-1}{\underbrace{1\dots 1}}\right],A=(1,\dots ,1)\in {\mathbb{F}}_p^{1\times (n-1)}.$$

Then ${\mathcal{C}}_{\mathrm{rep}}$ is a right-linear code over $\mathrm{NF}\left(p^2\right)$. Moreover, it is left-stable by Theorem 4.

4. Hermitian Dickson Inner Product and LCD Codes

In this section, we investigate Hermitian Dickson LCD codes over the Dickson near-field $\mathrm{NF}\left(p^2\right)$. Motivated by the classical theory of LCD codes over finite fields [17,18,19], we consider complementary-dual codes with respect to the Hermitian Dickson inner product and develop systematic criteria in the ${\mathbb{F}}_p$–systematic setting.

4.1. Hermitian Dickson Inner Product and Duality

The Hermitian Dickson inner product of $\mathbf{x}$ and $\mathbf{y}$ is defined by

$${\langle \mathbf{x},\mathbf{y}\rangle }_H:=\sum _{i=1}^n{x}_i\circ \overline{y_i}\in \mathrm{NF}\left(p^2\right).$$

For a right-linear code $\mathcal{C}\subseteq \mathcal{V}$, its Hermitian Dickson dual is

$${\mathcal{C}}^{{\perp }_H}:=\left\{\mathbf{v}\in \mathcal{V}:{\langle \mathbf{v},\mathbf{c}\rangle }_H=0\mathrm{for}\mathrm{all}\mathbf{c}\in \mathcal{C}\right\}.$$

We say that $\mathcal{C}$ is Hermitian Dickson self-orthogonal if $\mathcal{C}\subseteq {\mathcal{C}}^{{\perp }_H}$, and Hermitian Dickson self-dual if $\mathcal{C}={\mathcal{C}}^{{\perp }_H}$. Moreover, $\mathcal{C}$ is called a Hermitian Dickson LCD code if

$$\mathcal{C}\cap {\mathcal{C}}^{{\perp }_H}=\left\{\mathbf{0}\right\}.$$

We first record that ${\langle ·,·\rangle }_H$ is invariant under coordinate permutations. For $\pi \in S_n$, we let $\pi$ act on $\mathcal{V}$ by

$$\pi (x_1,\dots ,x_n):=(x_{{\pi }^{-1}\left(1\right)},\dots ,x_{{\pi }^{-1}\left(n\right)}).$$

Lemma 16.

For any $\pi \in S_n$ and any $\mathbf{x},\mathbf{y}\in \mathcal{V}$,

$${\langle \pi \left(\mathbf{x}\right),\pi \left(\mathbf{y}\right)\rangle }_H={\langle \mathbf{x},\mathbf{y}\rangle }_H.$$

Consequently, for any right-linear code $\mathcal{C}$ and any $\pi \in S_n$,

$$\pi \left({\mathcal{C}}^{{\perp }_H}\right)=\pi {\left(\mathcal{C}\right)}^{{\perp }_H}.$$

Proof. 

The first identity follows by a change of index in the defining sum:

$${\langle \pi \left(\mathbf{x}\right),\pi \left(\mathbf{y}\right)\rangle }_H=\sum _{i=1}^n{x}_{{\pi }^{-1}\left(i\right)}\circ \overline{y_{{\pi }^{-1}\left(i\right)}}=\sum _{j=1}^n{x}_j\circ \overline{y_j}={\langle \mathbf{x},\mathbf{y}\rangle }_H.$$

For the consequence, let $\mathbf{v}\in \mathcal{V}$ and set $\mathbf{z}=\pi \left(\mathbf{v}\right)$. For any $\mathbf{c}\in \mathcal{C}$, the first part gives

$${\langle \mathbf{z},\pi \left(\mathbf{c}\right)\rangle }_H={\langle \pi \left(\mathbf{v}\right),\pi \left(\mathbf{c}\right)\rangle }_H={\langle \mathbf{v},\mathbf{c}\rangle }_H.$$

Thus ${\langle \mathbf{v},\mathbf{c}\rangle }_H=0$ for all $\mathbf{c}\in \mathcal{C}$ if and only if ${\langle \pi \left(\mathbf{v}\right),{\mathbf{c}}^{\prime }\rangle }_H=0$ for all ${\mathbf{c}}^{\prime }\in \pi \left(\mathcal{C}\right)$, which proves the claim.   □

We now record the basic identities of the Hermitian Dickson inner product. In particular, ${\langle ·,·\rangle }_H$ is ${\mathbb{F}}_p$–linear in the first argument and compatible with the coordinate-wise right $\mathrm{NF}\left(p^2\right)$–action in the second.

Lemma 17.

For all $\mathbf{x},\mathbf{y},\mathbf{z}\in \mathrm{NF}{\left(p^2\right)}^n$, $a\in \mathrm{NF}\left(p^2\right)$, and $\lambda \in {\mathbb{F}}_p$:

 (a) 

${\langle \mathbf{x}+\mathbf{z},\mathbf{y}\rangle }_H={\langle \mathbf{x},\mathbf{y}\rangle }_H+{\langle \mathbf{z},\mathbf{y}\rangle }_H$.

 (b) 

${\langle \lambda \mathbf{x},\mathbf{y}\rangle }_H=\lambda {\langle \mathbf{x},\mathbf{y}\rangle }_H$.

 (c) 

${\langle \mathbf{x},\mathbf{y}\circ a\rangle }_H={\langle \mathbf{x},\mathbf{y}\rangle }_H\circ \overline{a}$.

 (d) 

$\overline{{\langle \mathbf{x},\mathbf{y}\rangle }_H}={\sum }_{i=1}^n\overline{x_i}\circ y_i$.

 (e) 

${〈\overline{\mathbf{x}},\overline{\mathbf{y}}〉}_H=\overline{{〈\mathbf{x},\mathbf{y}〉}_H}$.

 (f) 

If ${\langle \mathbf{x},\mathbf{y}\rangle }_H=0$ for all $\mathbf{y}\in \mathrm{NF}{\left(p^2\right)}^n$, then $\mathbf{x}=\mathbf{0}$. Likewise, if ${\langle \mathbf{x},\mathbf{y}\rangle }_H=0$ for all $\mathbf{x}\in \mathrm{NF}{\left(p^2\right)}^n$, then $\mathbf{y}=\mathbf{0}$.

Proof. 

Write $\mathbf{x}=(x_1,\dots ,x_n)$ and $\mathbf{y}=(y_1,\dots ,y_n)$.

(a)

Using right distributivity in Lemma 1(a) coordinate-wise,

$${\langle \mathbf{x}+\mathbf{z},\mathbf{y}\rangle }_H=\sum _{i=1}^n(x_i+z_i)\circ \overline{y_i}=\sum _{i=1}^n\left(x_i\circ \overline{y_i}+z_i\circ \overline{y_i}\right)={\langle \mathbf{x},\mathbf{y}\rangle }_H+{\langle \mathbf{z},\mathbf{y}\rangle }_H.$$

(b)

Let $\lambda \in {\mathbb{F}}_p$. For each i, the map $t\mapsto t\circ \overline{y_i}$ is ${\mathbb{F}}_p$–linear by Lemma 2 (applied with $n=1$), hence

$${\langle \lambda \mathbf{x},\mathbf{y}\rangle }_H=\sum _{i=1}^n\left(\lambda x_i\right)\circ \overline{y_i}=\lambda \sum _{i=1}^n{x}_i\circ \overline{y_i}=\lambda {\langle \mathbf{x},\mathbf{y}\rangle }_H.$$

(c)

Since $\mathbf{y}\circ a=(y_1\circ a,\dots ,y_n\circ a)$, we have

$${\langle \mathbf{x},\mathbf{y}\circ a\rangle }_H=\sum _{i=1}^n{x}_i\circ \overline{y_i\circ a}.$$

By Lemma 4(d), $\overline{y_i\circ a}=\overline{y_i}\circ \overline{a}$, and therefore, by associativity,

$$x_i\circ \overline{y_i\circ a}=x_i\circ (\overline{y_i}\circ \overline{a})=(x_i\circ \overline{y_i})\circ \overline{a}.$$

Applying right distributivity in the right factor $\overline{a}$ yields

$${\langle \mathbf{x},\mathbf{y}\circ a\rangle }_H=\sum _{i=1}^n(x_i\circ \overline{y_i})\circ \overline{a}=\left(\sum _{i=1}^n{x}_i\circ \overline{y_i}\right)\circ \overline{a}={\langle \mathbf{x},\mathbf{y}\rangle }_H\circ \overline{a}.$$

(d)

Using ${\mathbb{F}}_p$–linearity of conjugation in Lemma 4(a) and multiplicativity with respect to ∘ in Lemma 4(d),

$$\overline{{\langle \mathbf{x},\mathbf{y}\rangle }_H}=\overline{\sum _{i=1}^n{x}_i\circ \overline{y_i}}=\sum _{i=1}^n\overline{x_i\circ \overline{y_i}}=\sum _{i=1}^n\overline{x_i}\circ \overline{\overline{y_i}}=\sum _{i=1}^n\overline{x_i}\circ y_i.$$

(e)

By definition and Lemma 4(b),

$${〈\overline{\mathbf{x}},\overline{\mathbf{y}}〉}_H=\sum _{i=1}^n\overline{x_i}\circ \overline{\overline{y_i}}=\sum _{i=1}^n\overline{x_i}\circ y_i=\overline{{\langle \mathbf{x},\mathbf{y}\rangle }_H},$$

where the last equality is exactly (d).

(f)

Suppose ${\langle \mathbf{x},\mathbf{y}\rangle }_H=0$ for all $\mathbf{y}\in \mathrm{NF}{\left(p^2\right)}^n$. If $\mathbf{x}\ne \mathbf{0}$, choose j with $x_j\ne 0$ and take $\mathbf{y}=e_j$ the jth standard basis vector. Then $0={\langle \mathbf{x},e_j\rangle }_H=x_j\circ \overline{1}=x_j\circ 1=x_j$, a contradiction. Hence, $\mathbf{x}=\mathbf{0}$.

The second assertion is proved similarly: if ${\langle \mathbf{x},\mathbf{y}\rangle }_H=0$ for all $\mathbf{x}$, then taking $\mathbf{x}=e_j$ gives $0={\langle e_j,\mathbf{y}\rangle }_H=1\circ \overline{y_j}=\overline{y_j}$, so $y_j=0$ for all j and thus $\mathbf{y}=\mathbf{0}$.

□

Corollary 10.

For any right-linear code $\mathcal{C}$ one has

$$\overline{{\mathcal{C}}^{{\perp }_H}}={\left(\overline{\mathcal{C}}\right)}^{{\perp }_H}.$$

Proof. 

Let $\mathbf{w}\in \overline{{\mathcal{C}}^{{\perp }_H}}$. Then $\mathbf{w}=\overline{\mathbf{v}}$ for some $\mathbf{v}\in {\mathcal{C}}^{{\perp }_H}$. Take any $\mathbf{u}\in \overline{\mathcal{C}}$, so $\mathbf{u}=\overline{\mathbf{c}}$ for some $\mathbf{c}\in \mathcal{C}$. By Lemma 17(e),

$${\langle \mathbf{w},\mathbf{u}\rangle }_H={〈\overline{\mathbf{v}},\overline{\mathbf{c}}〉}_H=\overline{{\langle \mathbf{v},\mathbf{c}\rangle }_H}=0,$$

hence $\mathbf{w}\in {\left(\overline{\mathcal{C}}\right)}^{{\perp }_H}$. This shows $\overline{{\mathcal{C}}^{{\perp }_H}}\subseteq {\left(\overline{\mathcal{C}}\right)}^{{\perp }_H}$.

Conversely, let $\mathbf{w}\in {\left(\overline{\mathcal{C}}\right)}^{{\perp }_H}$ and set $\mathbf{v}=\overline{\mathbf{w}}$. For any $\mathbf{c}\in \mathcal{C}$ we have $\overline{\mathbf{c}}\in \overline{\mathcal{C}}$, hence ${\langle \mathbf{w},\overline{\mathbf{c}}\rangle }_H=0$. Taking Dickson conjugates and using Lemma 17(e),

$${\langle \mathbf{v},\mathbf{c}\rangle }_H=\overline{{\langle \mathbf{w},\overline{\mathbf{c}}\rangle }_H}=0.$$

Thus $\mathbf{v}\in {\mathcal{C}}^{{\perp }_H}$, so $\mathbf{w}=\overline{\mathbf{v}}\in \overline{{\mathcal{C}}^{{\perp }_H}}$. Therefore ${\left(\overline{\mathcal{C}}\right)}^{{\perp }_H}\subseteq \overline{{\mathcal{C}}^{{\perp }_H}}$, and equality follows.   □

Remark 5.

Lemma 17 is inherently one-sided: parts (a) and (b) concern additivity and ${\mathbb{F}}_p$–linearity in the first argument. In general there is no corresponding additivity (or ${\mathbb{F}}_p$–linearity) in the second argument, because the Dickson product is not left distributive. Likewise, the inner product is not right $\mathrm{NF}\left(p^2\right)$–linear in the second variable; the most one can expect is the $\overline{(·)}$–semilinearity recorded in Lemma 17(c). The following computations make both failures explicit.

For concreteness, work in $\mathrm{NF}\left(9\right)$ with underlying field ${\mathbb{F}}_9={\mathbb{F}}_3\left(\alpha \right)$, where ${\alpha }^2=2$, and Dickson conjugation $\overline{t}=t^3$. Recall that $\overline{\alpha }={\alpha }^3=2\alpha$ and that for $b\notin H_0$ one has $u\circ b=u^{3}b$.

 (i) 

Failure of additivity in the second argument. Take $n=1$, $x=\alpha$, $y=\alpha$, and $z=1$. Then $\overline{\alpha +1}={(\alpha +1)}^3=1+2\alpha$, and hence

$${\langle \alpha ,\alpha +1\rangle }_H=\alpha \circ \overline{\alpha +1}=\alpha \circ (1+2\alpha ).$$

Since $1+2\alpha \notin H_0$, we have

$$\alpha \circ (1+2\alpha )={\alpha }^3(1+2\alpha )=2\alpha (1+2\alpha )=2+2\alpha .$$

On the other hand,

$${\langle \alpha ,\alpha \rangle }_H+{\langle \alpha ,1\rangle }_H=\alpha \circ \overline{\alpha }+\alpha \circ \overline{1}=\alpha \circ \left(2\alpha \right)+\alpha =1+\alpha ,$$

because $2\alpha \in H_0$ and $\alpha \circ \left(2\alpha \right)=\alpha \left(2\alpha \right)=2{\alpha }^2=1$ in ${\mathbb{F}}_9$. Thus

$${\langle \alpha ,\alpha +1\rangle }_H\ne {\langle \alpha ,\alpha \rangle }_H+{\langle \alpha ,1\rangle }_H.$$

 (iI) 

Failure of right $\mathrm{NF}\left(p^2\right)$–linearity in the second argument. Still with $n=1$, take $x=y=1$ and $a=\alpha \notin {\mathbb{F}}_3$, so $\overline{a}=2\alpha \ne a$. Then

$${\langle 1,1\circ a\rangle }_H={\langle 1,\alpha \rangle }_H=1\circ \overline{\alpha }=1\circ \left(2\alpha \right)=2\alpha ,{\langle 1,1\rangle }_H\circ a=(1\circ \overline{1})\circ \alpha =\alpha .$$

Hence, ${\langle 1,1\circ a\rangle }_H\ne {\langle 1,1\rangle }_H\circ a$. At the same time, Lemma 17(c) holds sharply:

$${\langle 1,1\circ a\rangle }_H={\langle 1,1\rangle }_H\circ \overline{a}.$$

Corollary 11.

Let $\mathcal{C}$ be a right-linear code. Then:

 (a) 

$\mathcal{C}$ is Hermitian Dickson self-orthogonal if and only if $\overline{\mathcal{C}}$ is Hermitian Dickson self-orthogonal.

 (b) 

$\mathcal{C}$ is Hermitian Dickson self-dual if and only if $\overline{\mathcal{C}}$ is Hermitian Dickson self-dual.

Proof. 

By Corollary 10, $\overline{{\mathcal{C}}^{{\perp }_H}}={\left(\overline{\mathcal{C}}\right)}^{{\perp }_H}$. Conjugating the inclusion $\mathcal{C}\subseteq {\mathcal{C}}^{{\perp }_H}$ and the equality $\mathcal{C}={\mathcal{C}}^{{\perp }_H}$ and using $\overline{\overline{\mathcal{C}}}=\mathcal{C}$ yields both claims.   □

Corollary 12.

For any right-linear code $\mathcal{C}$ of length n, the code $\mathcal{C}$ is Hermitian Dickson LCD if and only if $\overline{\mathcal{C}}$ is Hermitian Dickson LCD.

Proof. 

Since Galois conjugation is a bijection on $\mathrm{NF}{\left(p^2\right)}^n$, it preserves intersections and $\left\{\mathbf{0}\right\}$. Using Corollary 10,

$$\overline{\mathcal{C}\cap {\mathcal{C}}^{{\perp }_H}}=\overline{\mathcal{C}}\cap \overline{{\mathcal{C}}^{{\perp }_H}}=\overline{\mathcal{C}}\cap {\left(\overline{\mathcal{C}}\right)}^{{\perp }_H}.$$

Thus $\mathcal{C}\cap {\mathcal{C}}^{{\perp }_H}=\left\{\mathbf{0}\right\}$ if and only if $\overline{\mathcal{C}}\cap {\left(\overline{\mathcal{C}}\right)}^{{\perp }_H}=\left\{\mathbf{0}\right\}$.   □

Corollary 13.

For any permutation $\pi \in S_n$ and any right-linear code $\mathcal{C}$:

 (a) 

$\mathcal{C}$ is Hermitian Dickson self-orthogonal if and only if $\pi \left(\mathcal{C}\right)$ is.

 (b) 

$\mathcal{C}$ is Hermitian Dickson self-dual if and only if $\pi \left(\mathcal{C}\right)$ is.

 (c) 

$\mathcal{C}$ is Hermitian Dickson LCD if and only if $\pi \left(\mathcal{C}\right)$ is.

Proof. 

Lemma 16 yields $\pi \left({\mathcal{C}}^{{\perp }_H}\right)=\pi {\left(\mathcal{C}\right)}^{{\perp }_H}$. Since $\pi$ is a bijection, $\pi (\mathcal{C}\cap {\mathcal{C}}^{{\perp }_H})=\pi \left(\mathcal{C}\right)\cap \pi \left({\mathcal{C}}^{{\perp }_H}\right)$. Applying $\pi$ to $\mathcal{C}\subseteq {\mathcal{C}}^{{\perp }_H}$, $\mathcal{C}={\mathcal{C}}^{{\perp }_H}$, and $\mathcal{C}\cap {\mathcal{C}}^{{\perp }_H}=\left\{\mathbf{0}\right\}$ gives (a)–(c).   □

4.2. Duals of Systematic Right-Linear Codes

We now give a constructive description of the Hermitian Dickson dual of a systematic right-linear code. Because $\mathrm{NF}\left(p^2\right)$ is only right distributive, the set ${\mathcal{C}}^{{\perp }_H}$ need not be a right near-subspace a priori. Under a natural stability hypothesis on the conjugate code, we prove that ${\mathcal{C}}^{{\perp }_H}$ is again a right-linear code and contains the right row span of a canonical systematic matrix. In the ${\mathbb{F}}_p$–systematic case this matrix generates ${\mathcal{C}}^{{\perp }_H}$ and yields the expected dimension relation.

Theorem 5.

Let $\mathcal{C}$ be a right-linear code of right dimension k with systematic generator matrix $G=\left[I_{k}A\right]$, where $A\in \mathrm{NF}{\left(p^2\right)}^{k\times (n-k)}$. Set $H=\left[-{\overline{A}}^{\top }{I}_{n-k}\right]\in \mathrm{NF}{\left(p^2\right)}^{(n-k)\times n}$, and let $\langle H\rangle$ denote the right row span of H. Assume that $\overline{\mathcal{C}}$ is left-stable. Then:

 (i) 

${\mathcal{C}}^{{\perp }_H}$ is a right-linear code.

 (ii) 

$\langle H\rangle \subseteq {\mathcal{C}}^{{\perp }_H}$.

 (iii) 

In particular, ${dim}_R\left({\mathcal{C}}^{{\perp }_H}\right)\ge n-k$, and hence ${dim}_R\left(\mathcal{C}\right)+{dim}_R\left({\mathcal{C}}^{{\perp }_H}\right)\ge n$.

Proof. 

(i)

The set ${\mathcal{C}}^{{\perp }_H}$ is an additive subgroup of $\mathcal{V}$ by Lemma 17(a). Let $\mathbf{v}\in {\mathcal{C}}^{{\perp }_H}$ and $a\in \mathrm{NF}\left(p^2\right)$. Fix $\mathbf{c}\in \mathcal{C}$ and set $\mathbf{u}=\overline{\mathbf{c}}\in \overline{\mathcal{C}}$. By hypothesis, $a\circ \mathbf{u}\in \overline{\mathcal{C}}$, hence there exists ${\mathbf{c}}^{\prime }\in \mathcal{C}$ such that $a\circ \overline{\mathbf{c}}=\overline{{\mathbf{c}}^{\prime }}$. Using associativity of ∘,

$${\langle \mathbf{v}\circ a,\mathbf{c}\rangle }_H=\sum _{i=1}^n(v_i\circ a)\circ \overline{c_i}=\sum _{i=1}^n{v}_i\circ \left(a\circ \overline{c_i}\right)=\sum _{i=1}^n{v}_i\circ \overline{c_i^{\prime }}={\langle \mathbf{v},{\mathbf{c}}^{\prime }\rangle }_H=0.$$

Thus $\mathbf{v}\circ a\in {\mathcal{C}}^{{\perp }_H}$, proving that ${\mathcal{C}}^{{\perp }_H}$ is a right-linear code.

(ii)

Let ${\mathbf{g}}_1,\dots ,{\mathbf{g}}_k$ be the rows of G. Any $\mathbf{c}\in \mathcal{C}$ can be written as $\mathbf{c}={\sum }_{i=1}^k{\mathbf{g}}_i\circ u_i$ for some $u_1,\dots ,u_k\in \mathrm{NF}\left(p^2\right)$, hence $\mathbf{c}=\left(\mathbf{u}\right|\mathbf{u}A)$ with $\mathbf{u}=(u_1,\dots ,u_k)$ and

$${\left(\mathbf{u}A\right)}_j=\sum _{i=1}^k{A}_{i,j}\circ u_i(j=1,\dots ,n-k).$$

Let ${\mathbf{h}}_1,\dots ,{\mathbf{h}}_{n-k}$ be the rows of H. For $j\in \{1,\dots ,n-k\}$,

$${\mathbf{h}}_j=\left(-\overline{A_{1,j}},\dots ,-\overline{A_{k,j}}|e_j\right).$$

By additivity of conjugation and $\overline{x\circ y}=\overline{x}\circ \overline{y}$,

$$\overline{{\left(\mathbf{u}A\right)}_j}=\overline{\sum _{i=1}^k{A}_{i,j}\circ u_i}=\sum _{i=1}^k\overline{A_{i,j}}\circ \overline{u_i}.$$

Therefore

$${\langle {\mathbf{h}}_j,\mathbf{c}\rangle }_H=\sum _{i=1}^k(-\overline{A_{i,j}})\circ \overline{u_i}+1\circ \overline{{\left(\mathbf{u}A\right)}_j}=0,$$

so ${\mathbf{h}}_j\in {\mathcal{C}}^{{\perp }_H}$ for all j. Hence, $\langle H\rangle \subseteq {\mathcal{C}}^{{\perp }_H}$.

(iii)

Suppose ${\sum }_{j=1}^{n-k}{\mathbf{h}}_j\circ a_j=\mathbf{0}$. Looking at the last $n-k$ coordinates yields $(a_1,\dots ,a_{n-k})=\mathbf{0}$, hence $a_j=0$ for all j. Thus the rows of H are right independent and ${dim}_R\left(\langle H\rangle \right)=n-k$. Using (ii) gives ${dim}_R\left({\mathcal{C}}^{{\perp }_H}\right)\ge n-k$, and the final inequality follows.

□

Corollary 14.

In the setting of Theorem 5, assume in addition that $A_{i,j}\in {\mathbb{F}}_p$ for all $i,j$. Then:

 (i) 

${\mathcal{C}}^{{\perp }_H}$ is a right-linear code.

 (ii) 

${\mathcal{C}}^{{\perp }_H}=\langle H\rangle$.

 (iii) 

In particular, ${dim}_R\left({\mathcal{C}}^{{\perp }_H}\right)=n-k$, and hence

$${dim}_R\left(\mathcal{C}\right)+{dim}_R\left({\mathcal{C}}^{{\perp }_H}\right)=n.$$

Proof. 

Statement (i) is Theorem 5(i), and $\langle H\rangle \subseteq {\mathcal{C}}^{{\perp }_H}$ holds by Theorem 5(ii).

For the reverse inclusion, let $\mathbf{v}=\left(\mathbf{p}\right|\mathbf{q})\in {\mathcal{C}}^{{\perp }_H}$ with $\mathbf{p}\in \mathrm{NF}{\left(p^2\right)}^k$ and $\mathbf{q}\in \mathrm{NF}{\left(p^2\right)}^{n-k}$. For each $i\in \{1,\dots ,k\}$, since ${\mathbf{g}}_i=\left(e_i\right|A_{i,\ast })\in \mathcal{C}$, we have

$$0={\langle \mathbf{v},{\mathbf{g}}_i\rangle }_H=p_i+\sum _{j=1}^{n-k}{q}_j\circ \overline{A_{i,j}}.$$

Because $A_{i,j}\in {\mathbb{F}}_p$, we have $\overline{A_{i,j}}=A_{i,j}$, and elements of ${\mathbb{F}}_p$ are central in $\mathrm{NF}\left(p^2\right)$. Therefore

$$p_i=\sum _{j=1}^{n-k}(-A_{i,j})\circ q_j.$$

Let ${\mathbf{h}}_1,\dots ,{\mathbf{h}}_{n-k}$ be the rows of $H=[-A^{\top }{I}_{n-k}]$. The identities above imply

$$\mathbf{v}=\sum _{j=1}^{n-k}{\mathbf{h}}_j\circ q_j\in \langle H\rangle ,$$

so ${\mathcal{C}}^{{\perp }_H}\subseteq \langle H\rangle$. This proves (ii), and (iii) follows from ${dim}_R\left(\langle H\rangle \right)=n-k$ as in Theorem 5(iii).   □

Corollary 15.

Assume the hypotheses of Theorem 5. Then:

(i)

If $\mathcal{C}$ is Hermitian Dickson self-dual, then ${dim}_R\left(\mathcal{C}\right)\ge {\displaystyle \frac{n}{2}}$.

(ii)

If $\mathcal{C}$ is Hermitian Dickson self-orthogonal, then ${dim}_R\left({\mathcal{C}}^{{\perp }_H}\right)\ge {\displaystyle \frac{n}{2}}$.

Proof. 

By Theorem 5(iii), ${dim}_R\left(\mathcal{C}\right)+{dim}_R\left({\mathcal{C}}^{{\perp }_H}\right)\ge n$. If $\mathcal{C}={\mathcal{C}}^{{\perp }_H}$, then $2{dim}_R\left(\mathcal{C}\right)\ge n$, proving (i). If $\mathcal{C}\subseteq {\mathcal{C}}^{{\perp }_H}$, then ${dim}_R\left(\mathcal{C}\right)\le {dim}_R\left({\mathcal{C}}^{{\perp }_H}\right)$ and $n\le 2{dim}_R\left({\mathcal{C}}^{{\perp }_H}\right)$, proving (ii).   □

Corollary 16.

 Assume the hypotheses of Corollary 14. Then: 

(i)

If $\mathcal{C}$ is Hermitian Dickson self-orthogonal, then ${dim}_R\left(\mathcal{C}\right)\le {\displaystyle \frac{n}{2}}$.

(ii)

If $\mathcal{C}$ is Hermitian Dickson self-dual, then n is even and ${dim}_R\left(\mathcal{C}\right)={\displaystyle \frac{n}{2}}$.

Proof. 

By Corollary 14(iii), ${dim}_R\left(\mathcal{C}\right)+{dim}_R\left({\mathcal{C}}^{{\perp }_H}\right)=n$. If $\mathcal{C}\subseteq {\mathcal{C}}^{{\perp }_H}$, then ${dim}_R\left(\mathcal{C}\right)\le {dim}_R$, hence $n\ge 2{dim}_R\left(\mathcal{C}\right)$, proving (i). If $\mathcal{C}={\mathcal{C}}^{{\perp }_H}$, then $n=2{dim}_R\left(\mathcal{C}\right)$, proving (ii).   □

Corollary 17.

Under the hypotheses of Corollary 14, the standard size relation holds:

$$\left|\mathcal{C}\right|·\left|{\mathcal{C}}^{{\perp }_H}\right|={\left|\mathrm{NF}\left(p^2\right)\right|}^n=p^{2n}.$$

Proof. 

By Corollary 14(iii) and Corollary 4, we have $\left|\mathcal{C}\right|=|\mathrm{NF}\left(p^2\right){|}^k=p^{2k}$ and $|{\mathcal{C}}^{{\perp }_H}|=|\mathrm{NF}\left(p^2\right){|}^{n-k}=p^{2(n-k)}$. Therefore $\left|\mathcal{C}\right|·|{\mathcal{C}}^{{\perp }_H}|=p^{2k}·p^{2(n-k)}=p^{2n}={\left|\mathrm{NF}\left(p^2\right)\right|}^n$.   □

4.3. Gram–Type Criteria in the ${\mathbb{F}}_p$–Systematic Case

We now restrict to the ${\mathbb{F}}_p$–systematic case and assume the hypotheses of Corollary 14. In this regime, Galois conjugation fixes ${\mathbb{F}}_p$ (so $\overline{A}=A$) and ${\mathbb{F}}_p\subseteq Z\left(\mathrm{NF}\left(p^2\right)\right)$, hence the Hermitian Dickson dual is generated by the canonical matrix

$$H=\left[-A^{\top }{I}_{n-k}\right].$$

Moreover, the matrix of Hermitian Dickson inner products between the rows of G coincides with the usual Gram matrix. Indeed, since $\overline{A}=A$, we have

$$G{\overline{G}}^{\top }=G{G}^{\top }=\left[I_{k}A\right]\left[\begin{array}{c}{I}_k\\ A^{\top }\end{array}\right]=I_k+A{A}^{\top }\in {\mathbb{F}}_p^{k\times k}.$$

Motivated by the classical Gram–matrix viewpoint in the study of LCD codes and hulls over finite fields (see, e.g., [17,20]), we define the Hermitian Dickson Gramian associated with G by $\Gamma :=I_k+A{A}^{\top }\in {\mathbb{F}}_p^{k\times k}$. For a matrix M over ${\mathbb{F}}_p$, we write ${rank}_{{\mathbb{F}}_p}\left(M\right)$ for its rank over ${\mathbb{F}}_p$ and set ${nullity}_{{\mathbb{F}}_p}\left(M\right):={dim}_{{\mathbb{F}}_p}{Ker}_{{\mathbb{F}}_p}\left(M\right).$

Proposition 5.

If ${\mathbf{g}}_1,\dots ,{\mathbf{g}}_k$ denote the rows of G, then ${\langle {\mathbf{g}}_r,{\mathbf{g}}_s\rangle }_H={\Gamma }_{r,s}$ for all $1\le r,s\le k$. In particular, $\mathcal{C}$ is Hermitian Dickson self-orthogonal if and only if $\Gamma =0$ in ${\mathbb{F}}_p^{k\times k}$.

Proof. 

Write ${\mathbf{g}}_r=\left(e_r\right|A_{r,\ast })$. Since $A\in {\mathbb{F}}_p$, we have $\overline{{\mathbf{g}}_s}={\mathbf{g}}_s$ and $x\circ c=xc$ for every $c\in {\mathbb{F}}_p$. Therefore

$${\langle {\mathbf{g}}_r,{\mathbf{g}}_s\rangle }_H=\sum _{i=1}^n{\left({\mathbf{g}}_r\right)}_i\circ \overline{{\left({\mathbf{g}}_s\right)}_i}=\sum _{i=1}^n{\left({\mathbf{g}}_r\right)}_i\circ {\left({\mathbf{g}}_s\right)}_i={\delta }_{rs}+\sum _{j=1}^{n-k}{A}_{rj}{A}_{sj},$$

which is exactly the $(r,s)$–entry of $I_k+A{A}^{\top }$, i.e., of $\Gamma$.

If $\Gamma =0$, then ${\langle {\mathbf{g}}_r,{\mathbf{g}}_s\rangle }_H=0$ for all $r,s$, hence each ${\mathbf{g}}_r\in {\mathcal{C}}^{{\perp }_H}$. Since $\mathcal{C}$ is generated from its rows by addition and the coordinate-wise right $\mathrm{NF}\left(p^2\right)$–action, it follows that $\mathcal{C}\subseteq {\mathcal{C}}^{{\perp }_H}$. Conversely, if $\mathcal{C}\subseteq {\mathcal{C}}^{{\perp }_H}$, then ${\langle {\mathbf{g}}_r,{\mathbf{g}}_s\rangle }_H=0$ for all $r,s$, hence $\Gamma =0$.   □

Proposition 6.

Let $\mathcal{C}\subseteq \mathrm{NF}{\left(p^2\right)}^n$ be a right-linear code generated by a systematic matrix $G=\left[I_{k}A\right]$ with $A\in {\mathbb{F}}_p^{k\times (n-k)}$, and let $\Gamma :=I_k+A{A}^{\top }\in {\mathbb{F}}_p^{k\times k}$ be the Hermitian Dickson Gramian. Then

$$\mathcal{C}\cap {\mathcal{C}}^{{\perp }_H}=\left\{\left(\mathbf{u}\right|\mathbf{u}A)\in \mathrm{NF}{\left(p^2\right)}^n:\mathbf{u}\in \mathrm{NF}{\left(p^2\right)}^k\mathrm{and}\mathbf{u}\Gamma =\mathbf{0}\right\}.$$

In particular, $\mathcal{C}$ is Hermitian Dickson LCD if and only if $\Gamma$ is nonsingular over ${\mathbb{F}}_p$.

Proof. 

By Corollary 14, ${\mathcal{C}}^{{\perp }_H}=\langle H\rangle$ with $H=[-A^{\top }{I}_{n-k}]$. Since G is systematic, every codeword of $\mathcal{C}$ is uniquely of the form $\left(\mathbf{u}\right|\mathbf{u}A)$ with $\mathbf{u}\in \mathrm{NF}{\left(p^2\right)}^k$.

Let ${\mathbf{h}}_1,\dots ,{\mathbf{h}}_{n-k}$ be the rows of H and let $\mathbf{q}=(q_1,\dots ,q_{n-k})\in \mathrm{NF}{\left(p^2\right)}^{n-k}$. A generic element of $\langle H\rangle$ is ${\sum }_{j=1}^{n-k}{\mathbf{h}}_j\circ q_j$. Because $A_{ij}\in {\mathbb{F}}_p$ is central in $\mathrm{NF}\left(p^2\right)$, every vector in $\langle H\rangle$ can be written in the form $(-\mathbf{q}{A}^{\top }|\mathbf{q})$, where ${\left(\mathbf{q}{A}^{\top }\right)}_i={\sum }_{j=1}^{n-k}{q}_j{A}_{ij}$.

Now let $\left(\mathbf{u}\right|\mathbf{u}A)\in \mathcal{C}\cap {\mathcal{C}}^{{\perp }_H}$. Then $\left(\mathbf{u}\right|\mathbf{u}A)\in \langle H\rangle$, so there exists $\mathbf{q}\in \mathrm{NF}{\left(p^2\right)}^{n-k}$ such that

$$\left(\mathbf{u}\right|\mathbf{u}A)=(-\mathbf{q}{A}^{\top }|\mathbf{q}).$$

Comparing the last $n-k$ coordinates yields $\mathbf{q}=\mathbf{u}A$, and substituting into the first k coordinates gives $\mathbf{u}=-\left(\mathbf{u}A\right)A^{\top }$. Since A has entries in ${\mathbb{F}}_p$, the product $\left(\mathbf{u}A\right)A^{\top }$ coincides with the usual matrix product $\mathbf{u}\left(A{A}^{\top }\right)$. Hence, $\mathbf{u}+\mathbf{u}\left(A{A}^{\top }\right)=\mathbf{0}$, equivalently $\mathbf{u}(I_k+A{A}^{\top })=\mathbf{0}$, i.e., $\mathbf{u}\Gamma =\mathbf{0}$. This proves the displayed characterization.

Conversely, if $\mathbf{u}\Gamma =\mathbf{0}$ and we set $\mathbf{q}:=\mathbf{u}A$, then $\mathbf{u}=-\mathbf{u}\left(A{A}^{\top }\right)=-\left(\mathbf{u}A\right)A^{\top }=-\mathbf{q}{A}^{\top }$, hence $\left(\mathbf{u}\right|\mathbf{u}A)=(-\mathbf{q}{A}^{\top }|\mathbf{q})\in \langle H\rangle ={\mathcal{C}}^{{\perp }_H}$. Therefore $\left(\mathbf{u}\right|\mathbf{u}A)\in \mathcal{C}\cap {\mathcal{C}}^{{\perp }_H}$.

For the LCD criterion, assume that $\Gamma$ is nonsingular over ${\mathbb{F}}_p$ and let $\mathbf{u}\in \mathrm{NF}{\left(p^2\right)}^k$ satisfy $\mathbf{u}\Gamma =\mathbf{0}$. Fix an ${\mathbb{F}}_p$–basis $\{1,\omega \}$ of the additive group of $\mathrm{NF}\left(p^2\right)$ and write $\mathbf{u}={\mathbf{u}}^{\prime }+\omega {\mathbf{u}}^{″}$ with ${\mathbf{u}}^{\prime },{\mathbf{u}}^{″}\in {\mathbb{F}}_p^k$. Since $\Gamma \in {\mathbb{F}}_p^{k\times k}$, the condition $\mathbf{u}\Gamma =\mathbf{0}$ implies ${\mathbf{u}}^{\prime }\Gamma =\mathbf{0}$ and ${\mathbf{u}}^{″}\Gamma =\mathbf{0}$ in ${\mathbb{F}}_p^k$, hence ${\mathbf{u}}^{\prime }=\mathbf{0}$ and ${\mathbf{u}}^{″}=\mathbf{0}$, and therefore $\mathbf{u}=\mathbf{0}$. Thus $\mathcal{C}\cap {\mathcal{C}}^{{\perp }_H}=\left\{\mathbf{0}\right\}$ and $\mathcal{C}$ is LCD.

Conversely, if $\Gamma$ is singular over ${\mathbb{F}}_p$, choose $0\ne \mathbf{t}\in {\mathbb{F}}_p^k$ with $\mathbf{t}\Gamma =\mathbf{0}$. Viewing $\mathbf{t}$ as an element of $\mathrm{NF}{\left(p^2\right)}^k$, the nonzero codeword $\left(\mathbf{t}\right|\mathbf{t}A)$ lies in $\mathcal{C}\cap {\mathcal{C}}^{{\perp }_H}$, so $\mathcal{C}$ is not LCD.   □

Corollary 18.

In the setting of Proposition 6,

$${dim}_R\left(\mathcal{C}\cap {\mathcal{C}}^{{\perp }_H}\right)={nullity}_{{\mathbb{F}}_p}(\Gamma )=k-{rank}_{{\mathbb{F}}_p}(\Gamma ).$$

Proof. 

By Proposition 6, the intersection is the image of the set $Ker(\Gamma ):=\{\mathbf{u}\in \mathrm{NF}{\left(p^2\right)}^k:\mathbf{u}\Gamma =\mathbf{0}\}$ under the injective map $\mathbf{u}\mapsto \left(\mathbf{u}\right|\mathbf{u}A)$, hence ${dim}_R(\mathcal{C}\cap {\mathcal{C}}^{{\perp }_H})={dim}_R(Ker(\Gamma ))$.

Since $\Gamma \in {\mathbb{F}}_p^{k\times k}$, the constraint $\mathbf{u}\Gamma =\mathbf{0}$ is ${\mathbb{F}}_p$–linear in the ${\mathbb{F}}_p$–coordinates of $\mathbf{u}$. Writing $\mathbf{u}={\mathbf{u}}^{\left(0\right)}+\omega {\mathbf{u}}^{\left(1\right)}$ with ${\mathbf{u}}^{\left(0\right)},{\mathbf{u}}^{\left(1\right)}\in {\mathbb{F}}_p^k$, we have $\mathbf{u}\Gamma =\mathbf{0}$ if and only if ${\mathbf{u}}^{\left(0\right)}\Gamma =\mathbf{0}$ and ${\mathbf{u}}^{\left(1\right)}\Gamma =\mathbf{0}$ in ${\mathbb{F}}_p^k$. Therefore $Ker(\Gamma )$ has ${\mathbb{F}}_p$–dimension $2{nullity}_{{\mathbb{F}}_p}(\Gamma )$.

On the other hand, $Ker(\Gamma )$ is a right near-subspace of $\mathrm{NF}{\left(p^2\right)}^k$, and $\mathrm{NF}\left(p^2\right)$ has ${\mathbb{F}}_p$–dimension 2, so the ${\mathbb{F}}_p$–dimension of any right near-subspace equals twice its right dimension. Hence ${dim}_R(Ker(\Gamma ))={nullity}_{{\mathbb{F}}_p}(\Gamma )$, and the stated identity follows.   □

Corollary 19.

Assume the hypotheses of Corollary 14, and let $G=\left[I_{k}A\right]$ generate $\mathcal{C}$ with $A\in {\mathbb{F}}_p^{k\times (n-k)}$. If $\mathcal{C}$ is Hermitian Dickson self-dual, then $n=2k$ and $I_k+A{A}^{\top }=0$.

Proof. 

Self-duality implies self-orthogonality, so Proposition 5 yields $I_k+A{A}^{\top }=0$. Moreover, Corollary 14(iii) gives $n={dim}_R\left(\mathcal{C}\right)+{dim}_R\left({\mathcal{C}}^{{\perp }_H}\right)=2k$, hence $n=2k$.   □

4.4. Hamming Distance via a Hermitian Dickson Parity-Check Matrix

We close the ${\mathbb{F}}_p$–systematic case with the standard parity-check characterization of the minimum Hamming distance. In this setting, the canonical matrix $H=[-\overline{A}{I}_{n-k}]$ serves as a Hermitian Dickson parity-check matrix for $\mathcal{C}$.

Theorem 6.

Assume the hypotheses of Corollary 14. Let $\mathcal{C}$ be a right-linear code generated by $G=\left[I_{k}A\right]$ with $A\in {\mathbb{F}}_p^{k\times (n-k)}$. Let $H=[-{\overline{A}}^{\top }{I}_{n-k}]\in \mathrm{NF}{\left(p^2\right)}^{(n-k)\times n}$, and write $h^{\left(1\right)},\dots ,h^{\left(n\right)}$ for the columns of H. Then $x\in \mathcal{C}$ if and only if $H\circ {\overline{x}}^{\top }=0$, equivalently

$$H\circ {\overline{x}}^{\top }=\sum _{j=1}^n{h}^{\left(j\right)}\circ \overline{x_j}=0.$$

Moreover, $d_H\left(\mathcal{C}\right)$ equals the smallest integer t for which there exist distinct indices $j_1,\dots ,j_t$ and elements $c_1,\dots ,c_t\in \mathrm{NF}{\left(p^2\right)}^{\times }$ such that $h^{\left(j_1\right)}\circ c_1+\dots +h^{\left(j_t\right)}\circ c_t=0$.

Equivalently, $d_H\left(\mathcal{C}\right)$ is the smallest number of columns of H that are right-linearly dependent over $\mathrm{NF}\left(p^2\right)$.

Proof. 

By Corollary 14, the rows of H generate ${\mathcal{C}}^{{\perp }_H}$; hence $x\in \mathcal{C}$ implies $H\circ {\overline{x}}^{\top }=0$.

Conversely, assume $H\circ {\overline{x}}^{\top }=0$ and write $x=\left(\mathbf{b}\right|\mathbf{d})$ with $\mathbf{b}\in \mathrm{NF}{\left(p^2\right)}^k$ and $\mathbf{d}\in \mathrm{NF}{\left(p^2\right)}^{n-k}$. Reading the equation row-wise gives, for each $j=1,\dots ,n-k$,

$$\overline{d_j}=\sum _{i=1}^k{A}_{i,j}\circ \overline{b_i}.$$

Applying Galois conjugation and using $\overline{A_{i,j}}=A_{i,j}$ yields $d_j={\sum }_{i=1}^k{A}_{i,j}\circ b_i$,

so $\mathbf{d}=\mathbf{b}A$ and therefore $x=\left(\mathbf{b}\right|\mathbf{b}A)\in \mathcal{C}$.

For the distance statement, observe that $H\circ {\overline{x}}^{\top }=0$ is equivalent to the column relation ${\sum }_{j=1}^n{h}^{\left(j\right)}\circ \overline{x_j}=0$. If $h^{\left(j_1\right)}\circ c_1+\cdots +h^{\left(j_t\right)}\circ c_t=0$ is a nontrivial right dependence among t columns, choose $x_{j_s}\in \mathrm{NF}\left(p^2\right)$ with $\overline{x_{j_s}}=c_s$ and set all other coordinates to 0. Then $H\circ {\overline{x}}^{\top }=0$, hence $x\in \mathcal{C}$, and ${\mathrm{wt}}_H\left(x\right)=t$, so $d_H\left(\mathcal{C}\right)\le t$.

Conversely, let $0\ne x\in \mathcal{C}$ and set $S=\{j:x_j\ne 0\}$. From ${\sum }_{j=1}^n{h}^{\left(j\right)}\circ \overline{x_j}=0$ we obtain the nontrivial dependence ${\sum }_{j\in S}{h}^{\left(j\right)}\circ \overline{x_j}=0$ among the columns indexed by S. Thus some $\left|S\right|={\mathrm{wt}}_H\left(x\right)$ columns are right-linearly dependent. Minimizing over nonzero $x\in \mathcal{C}$ gives the claim.   □

5. Numerical Section

In this section, we present a computational classification of Hermitian Dickson self-orthogonal right-linear codes over the Dickson near-field $\mathrm{NF}\left(p^2\right)$, for $p\in \{3,5,7\}$ and for short lengths ($n\le 8$) with right dimension $k\in \{1,2\}$. All computations and numerical data were obtained with the computer algebra system Magma [16], using implementations of the Dickson product ∘, Galois (Frobenius) conjugation, and the Hermitian Dickson inner product.

Throughout this section, code equivalence is taken with respect to right monomial equivalence, i.e., the action on $\mathrm{NF}{\left(p^2\right)}^n$ induced by monomial matrices.

For each admissible pair $(n,k)$, we first enumerate right-linear codes over $\mathrm{NF}\left(p^2\right)$ up to right monomial equivalence. Using Theorem 1, we then select a systematic representative $G=\left[I_{k}A\right]$; whenever possible, we choose an ${\mathbb{F}}_p$–systematic representative, meaning that $A\in {\mathbb{F}}_p^{k\times (n-k)}$. This ${\mathbb{F}}_p$–systematic choice is only a display convention when such a representative exists in the equivalence class; the enumeration itself is performed over all right-linear codes over $\mathrm{NF}\left(p^2\right)$ up to monomial equivalence. Hermitian Dickson self-orthogonality is tested via Proposition 5; in the ${\mathbb{F}}_p$–systematic case, this reduces to the condition $A{A}^{\top }=-I_k$. For each resulting code, we compute the minimum Hamming distance $d_H\left(\mathcal{C}\right)$ and the weight distribution. We also record whether $\mathcal{C}$ is Hermitian Dickson self-dual and whether it meets the Singleton bound (Theorem 3). Finally, we group the codes into right monomial equivalence classes and tabulate, for each class, a representative generator matrix together with the corresponding invariants.

We restrict our tabulation to codes for which the conjugate code $\overline{\mathcal{C}}$ is left-stable, so that Theorem 5 applies and the Hermitian right dual ${\mathcal{C}}^{{\perp }_H}$ is again right-linear. In the parameter ranges considered in

Table 1. Hermitian Dickson self-orthogonal codes over $\mathrm{NF}\left(9\right)$ for lengths $n\le 8$.

n	k	${\mathit{d}}_{\mathit{H}}$	A	$\mathbf{Aut}\left(\mathcal{C}\right)$	Weight Distribution (Magma)	Note
3	1	3	$\left[11\right]$	12	$[<0,1>,<3,8>]$	MDS
4	1	3	$\left[110\right]$	24	$[<0,1>,<3,8>]$
	2	3	$\left[\begin{array}{cc}1& 1\\ 1& 2\end{array}\right]$	48	$[<0,1>,<3,32>,<4,48>]$	Self-dual, MDS
5	1	3	$\left[1100\right]$	96	$[<0,1>,<3,8>]$
	2	3	$\left[\begin{array}{ccc}1& 1& 0\\ 1& 2& 0\end{array}\right]$	96	$[<0,1>,<3,32>,<4,48>]$
6	1	3	$\left[11000\right]$	576	$[<0,1>,<3,8>]$
		6	$\left[11111\right]$	1440	$[<0,1>,<6,8>]$	MDS
	2	3	$\left[\begin{array}{cccc}1& 1& 0& 0\\ 1& 2& 0& 0\end{array}\right]$	384	$[<0,1>,<3,32>,<4,48>]$
		3	$\left[\begin{array}{cccc}1& 0& 1& 0\\ 0& 1& 0& 1\end{array}\right]$	288	$[<0,1>,<3,16>,<6,64>]$
7	1	3	$\left[110000\right]$	4608	$[<0,1>,<3,8>]$
		6	$\left[111110\right]$	2880	$[<0,1>,<6,8>]$
	2	3	$\left[\begin{array}{ccccc}1& 1& 0& 0& 0\\ 1& 2& 0& 0& 0\end{array}\right]$	2304	$[<0,1>,<3,32>,<4,48>]$
		3	$\left[\begin{array}{ccccc}0& 0& 1& 1& 0\\ 1& 1& 0& 0& 0\end{array}\right]$	576	$[<0,1>,<3,16>,<6,64>]$
		3	$\left[\begin{array}{ccccc}0& 0& 0& 1& 1\\ 1& 1& 1& 1& 2\end{array}\right]$	288	$[<0,1>,<3,8>,<6,24>,$<7,48>]
8	1	3	$\left[1100000\right]$	46,080	$[<0,1>,<3,8>]$
		6	$\left[1111100\right]$	11,520	$[<0,1>,<6,8>]$
	2	3	$\left[\begin{array}{cccccc}1& 1& 0& 0& 0& 0\\ 1& 2& 1& 1& 1& 0\end{array}\right]$	576	$[<0,1>,<3,8>,<6,24>,<7,48>]$
		3	$\left[\begin{array}{cccccc}1& 1& 0& 0& 0& 0\\ 0& 0& 1& 1& 0& 0\end{array}\right]$	2304	$[<0,1>,<3,16>,<6,64>]$
		3	$\left[\begin{array}{cccccc}1& 1& 0& 0& 0& 0\\ 1& 2& 0& 0& 0& 0\end{array}\right]$	18,432	$[<0,1>,<3,32>,<4,48>]$
		6	$\left[\begin{array}{cccccc}1& 0& 1& 1& 1& 1\\ 0& 1& 1& 1& 2& 2\end{array}\right]$	768	$[<0,1>,<6,32>,<8,48>]$	$d_H=n-k$

,

Table 2. Hermitian Dickson self-orthogonal codes over $\mathrm{NF}\left(25\right)$ for lengths $n\le 8$.

n	k	${\mathit{d}}_{\mathit{H}}$	A	$\mathbf{Aut}\left(\mathcal{C}\right)$	Weight Distribution (Magma)	Note
2	1	2	$\left[2\right]$	8	$[<0,1>,<2,24>]$	Self-dual, MDS
3	1	2	$\left[20\right]$	32	$[<0,1>,<2,24>]$
4	1	2	$\left[200\right]$	256	$[<0,1>,<2,24>]$
	1	4	$\left[221\right]$	96	$[<0,1>,<4,24>]$	MDS
	1	4	$\left[123\right]$	96	$[<0,1>,<4,24>]$	MDS
	1	4	$\left[234\right]$	96	$[<0,1>,<4,24>]$	MDS
	2	2	$\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]$	128	$[<0,1>,<2,48>,<4,576>]$	$d_H=n-k$
5	1	5	$\left[1111\right]$	480	$[<0,1>,<5,24>]$	MDS
5	2	2	$\left[\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\end{array}\right]$	512	$[<0,1>,<2,48>,<4,576>]$
		4	$\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& 3\end{array}\right]$	80	$[<0,1>,<4,120>,<5,504>]$	MDS
6	1	6	$\left[22211\right]$	2880	$[<0,1>,<6,24>]$	MDS
	2	2	$\left[\begin{array}{cccc}0& 0& 0& 2\\ 0& 0& 2& 0\end{array}\right]$	4096	$[<0,1>,<2,48>,<4,576>]$
		2	$\left[\begin{array}{cccc}0& 0& 0& 2\\ 1& 2& 2& 0\end{array}\right]$	768	$[<0,1>,<2,24>,<4,24>,<6,576>]$
		4	$\left[\begin{array}{cccc}0& 1& 2& 2\\ 2& 1& 0& 2\end{array}\right]$	192	$[<0,1>,<4,72>,<6,552>]$	$d_H=n-k$
		4	$\left[\begin{array}{cccc}1& 1& 1& 1\\ 1& 1& 4& 4\end{array}\right]$	64	$[<0,1>,<4,48>,<5,48>,<6,528>]$	$d_H=n-k$
		4	$\left[\begin{array}{cccc}0& 1& 2& 2\\ 0& 2& 1& 3\end{array}\right]$	320	$[<0,1>,<4,120>,<5,504>]$	$d_H=n-k$
7	1	7	$\left[211111\right]$	20,160	$[<0,1>,<7,24>]$	MDS
	2	5	$\left[\begin{array}{ccccc}3& 4& 3& 3& 4\\ 3& 1& 1& 2& 2\end{array}\right]$	160	$[<0,1>,<5,24>,<6,120>,<7,480>]$	$d_H=n-k$
8	1	8	$\left[2222111\right]$	161,280	$[<0,1>,<8,24>]$	MDS
8	2	6	$\left[\begin{array}{cccccc}0& 1& 1& 2& 2& 2\\ 1& 0& 2& 2& 3& 4\end{array}\right]$	64	$[<0,1>,<6,72>,<7,48>,<8,504>]$	$d_H=n-k$

and

Table 3. Hermitian Dickson self-orthogonal codes over $\mathrm{NF}\left(49\right)$ lengths $3\le n\le 8$.

n	k	${\mathit{d}}_{\mathit{H}}$	A	$\mathbf{Aut}\left(\mathcal{C}\right)$	Weight Distribution (Magma)	Note
3	1	3	$\left[32\right]$	36	$[<0,1>,<3,48>]$	MDS
4	1	4	$\left[112\right]$	144	$[<0,1>,<4,48>]$	MDS
	2	3	$\left[\begin{array}{cc}3& 2\\ 5& 3\end{array}\right]$	72	$[<0,1>,<3,192>,<4,2208>]$	self-dual, MDS
5	1	5	$\left[1133\right]$	720	$[<0,1>,<5,48>]$	MDS
6	1	6	$\left[11113\right]$	4320	$[<0,1>,<6,48>]$	MDS
	2	3	$\left[\begin{array}{cccc}0& 0& 2& 3\\ 2& 3& 0& 0\end{array}\right]$	2592	$[<0,1>,<3,96>,<6,2304>]$
7	1	7	$\left[111111\right]$	30,240	$[<0,1>,<7,48>]$	MDS
	2	5	$\left[\begin{array}{ccccc}1& 1& 1& 1& 3\\ 3& 4& 1& 6& 0\end{array}\right]$	24	$[<0,1>,<5,48>,<6,240>,<7,2112>]$
8	1	8	$\left[2211111\right]$	241,920	$[<0,1>,<8,48>]$	MDS
8	2	7	$\left[\begin{array}{cccccc}1& 1& 1& 3& 2& 2\\ 1& 2& 3& 1& 1& 5\end{array}\right]$	2016	$[<0,1>,<7,384>,<8,2016>]$	MDS

, each such equivalence class admits an ${\mathbb{F}}_p$-systematic representative (hence the displayed matrices A lie in ${\mathbb{F}}_p$); this is a choice of normal form and does not mean the codes are over a field.

In the parameter ranges reported in Table 1, Table 2 and Table 3, we observe two notable patterns: (i) for each admissible pair $(n,k)$, only a small number of right monomial equivalence classes occur; (ii) within these ranges, the self-dual cases (when $n=2k$) coincide with the MDS cases, i.e., those attaining the Singleton bound.

Example 7.

Let $\mathrm{NF}\left(p^2\right)=\mathrm{NF}\left(25\right)$. Following the computational procedure described earlier in this section, we enumerate all admissible ${\mathbb{F}}_5$–systematic one-generator right-linear codes and classify them up to the same right monomial equivalence adopted throughout the tables. The computation shows that, for $(n,k)=(4,1)$, there are exactly four inequivalent classes. We present one ${\mathbb{F}}_5$–systematic generator for each class, together with the corresponding numerical invariants.

Among these, there is a unique non-MDS class, which may be represented by

$$G=\left[1200\right].$$

This class has $d_H\left(\mathcal{C}\right)=2$ with weight distribution $[<0,1>,<2,24>]$, and the automorphism-group size is $|Aut(\mathcal{C}\left)\right|=256$. Set

$$|M_4|:=4!(|{\mathbb{F}}_5^{\times }{\left|\right)}^4=24·4^4=6144,$$

the size of the right-monomial group acting on length 4. It follows that the corresponding equivalence class contains

$${\displaystyle \frac{|M_4|}{256}}=24$$

distinct codes that are right-monomially equivalent to the displayed representative.

The remaining three classes meet the Singleton bound and hence are MDS. They admit the following ${\mathbb{F}}_5$–systematic generators:

$$G=\left[1221\right],G=\left[1123\right],G=\left[1234\right].$$

Moreover, these three MDS classes share the same numerical invariants: $d_H\left(\mathcal{C}\right)=4$ and weight distribution $[<0,1>,<4,24>]$. For each of these three MDS classes, the automorphism-group size is $|Aut(\mathcal{C}\left)\right|=96$, and hence each corresponding equivalence class contains

$${\displaystyle \frac{|M_4|}{96}}=64$$

distinct codes that are right-monomially equivalent to the displayed representative. These three MDS codes are nevertheless pairwise inequivalent under right monomial equivalence.

Finally, we note that these results are in full agreement with the corresponding  Magma  computations, reproducing exactly the four entries listed under $n=4$ and $k=1$ in Table 2.

The complete classification results for $p=3,5,7$ are summarized in Table 1, Table 2 and Table 3, respectively, where each row corresponds to a right monomial equivalence class and reports the associated parameters and invariants.

6. Conclusions and Future Work

In this paper, we developed a systematic framework for right-linear codes over the Dickson near-field $\mathrm{NF}\left(p^2\right)$. We reviewed the Dickson near-field structure on ${\mathbb{F}}_{p^2}$ and the associated one-sided linear algebra, introduced right-linear codes as right near-subspaces of $\mathrm{NF}{\left(p^2\right)}^n$, and proved that, up to right monomial equivalence, every such code admits a systematic generator matrix. We established basic parameter relations, including a Singleton-type bound for the minimum Hamming distance, and showed that Galois conjugationdefines a Hamming isometry compatible with the equivalence notions considered. With respect to the Hermitian Dickson inner product, we developed the corresponding duality theory and examined Hermitian Dickson self-orthogonality, self-duality, and the LCD property in a setting where duals need not be right codes a priori. In the ${\mathbb{F}}_p$–systematic case, we obtained explicit parity-check descriptions and Gram-type criteria that enable effective verification of these duality properties.

Several directions for future work arise naturally. Two concrete directions appear particularly promising. First, motivated by the MDS instances observed in Table 1, Table 2 and Table 3, we plan to construct explicit infinite families of Hermitian Dickson MDS codes, including self-dual MDS codes in the square case $n=2k$. Second, we aim to relax the left-stability hypothesis used to ensure that the Hermitian right dual ${\mathcal{C}}^{{\perp }_H}$ remains right-linear, or to identify alternative sufficient conditions leading to parity-check-type descriptions of ${\mathcal{C}}^{{\perp }_H}$ for wider parameter ranges. First, extending the computational classification to larger lengths and higher right dimensions may reveal additional families of Hermitian Dickson self-dual and LCD codes with strong parameters. Second, it is of interest to move beyond the ${\mathbb{F}}_p$–systematic regime by developing computable criteria for when the parity block has entries in $\mathrm{NF}\left(p^2\right)$ and by identifying transparent sufficient conditions ensuring the left-stability hypotheses that control the behavior of Hermitian Dickson duals. Third, it would be worthwhile to generalize the theory to other Dickson near-fields and to suitable classes of near-rings (including non-unital settings) and to determine which components of the duality theory persist beyond the near-field case. Finally, it would be interesting to investigate MacWilliams-type relations and weight-enumerator phenomena adapted to one-sided algebraic structures and to develop decoding methods and explore potential applications of near-field codes.