Constructions and Enumerations of Self-Dual and LCD Double Circulant Codes over a Local Ring

Abstract

The construction of self-dual and linear complementary dual (LCD) codes over finite rings, particularly over semi-local and local structures, is an active area of research due to their algebraic richness and applications in communications and cryptography. In this paper, we investigate double circulant and double negacirculant codes over the local ring $R_{q,u,v}={\mathbb{F}}_q+u{\mathbb{F}}_q+v{\mathbb{F}}_q,u^2=v^2=uv=vu=0,$ where $q=p^m$ is an odd prime power. Unlike the semi-local case, where decomposition via non-trivial idempotents simplifies analysis, the local structure of $R_{q,u,v}$ (with only trivial idempotents) makes enumeration and classification significantly more challenging. We first establish necessary and sufficient conditions for such codes to be self-dual or LCD; we then count the solutions to key equations over ${\mathbb{F}}_q$, including $a{b}^q+b{a}^q=0,$ to enable their enumeration. We further show that Gray images preserve these properties, leading to good self-dual and LCD codes over ${\mathbb{F}}_q$, and present optimal examples over ${\mathbb{F}}_7$. Our results extend double circulant constructions to a new algebraic setting, providing both theoretical advancements and practically relevant code designs.

1. Introduction

In recent years, there has been a surge in interest in double circulant (DC) and double negacirculant (DNC) codes over rings, extending classical field-based constructions to richer algebraic settings. Much of this activity has focused on semi-local rings, such as ${\mathbb{F}}_q+u{\mathbb{F}}_q$ and ${\mathbb{F}}_q+u{\mathbb{F}}_q+v{\mathbb{F}}_q$, which admit non-trivial idempotent decompositions that enable elegant CRT analyses and lead to closed enumeration formulas for self-dual and LCD codes. Notable recent contributions along these lines include work on double circulant and double negacirculant codes over semi-local rings [1,2,3], as well as studies on broader non-local or chain ring families with specialized structures [4,5,6]. These developments have underscored both the structural richness of ring-linear code constructions and their effectiveness in producing good codes with strong parameters. More recent research has also expanded the range of ambient rings considered for Gray image analysis and asymptotic existence results, reflecting growing interest in the topic [7,8,9,10,11].

Despite these advances, several limitations remain in the current literature. Most recent studies on double circulant and double negacirculant codes over rings, such as those by Yadav et al. [12], Shi et al. [13], and Dinh et al. [14], have focused primarily on semi-local rings with convenient idempotent decompositions, where orthogonality conditions can be reduced to simple norm equations and enumeration becomes tractable. These results provide elegant asymptotic analyses but rely heavily on the algebraic simplicity of semi-local structures. In contrast, genuinely local non-chain rings, which lack non-trivial idempotents, have received very limited attention. In such settings, classical counting techniques no longer apply, and one must instead handle more involved orthogonality relations such as $a{b}^q+b{a}^q=0,$ which substantially complicates enumeration and structure theorems. Partial progress has been made for double circulant codes over chain and non-local rings [8,9,15,16,17] and for related non-trivial extensions [12,13,14], but the double negacirculant case over local rings remains essentially unexplored, with existing work often restricted to special lengths or relying on reductions to the field case. Moreover, much of the literature applies field-level factorization results to rings with nilpotent elements without explicitly establishing comaximality or irreducibility at the ring level—an essential step for the rigorous application of the Chinese Remainder Theorem. Duality and LCD conditions under Gray maps are also frequently stated without clearly specifying the induced inner product or block length, making it difficult to rigorously verify the preservation of orthogonality. Finally, existing asymptotic results often give only qualitative existence statements, with key constants left implicit in the union-bound derivations; see, for instance [4,12,13,18], where the contribution of the non-constant CRT components is not fully isolated.

The present work aims to address these limitations by developing a systematic theory of self-dual and LCD double circulant and double negacirculant codes over the local, non-chain ring

$$R_{q,u,v}={\mathbb{F}}_q+u{\mathbb{F}}_q+v{\mathbb{F}}_q,u^2=v^2=uv=vu=0,$$

(1)

a setting that has attracted increased interest but remains far less developed in the context of DC and DNC codes. Our approach provides a fully rigorous treatment of CRT factorization and Hensel lifting in the presence of nilpotents, ensuring the validity of the decompositions used in the sequel. By treating both circulant (odd n) and negacirculant (even n) cases in parallel, we expand the range of known code families beyond those based on semi-local rings. We also give precise duality and LCD characterizations with explicit ambient rings and unit conditions, and we clarify the exact role of the Gray map in preserving inner products. Finally, our asymptotic analysis is derived with all constants made explicit, using the non-constant Gray block of length $3n$ to produce sharp and rigorous distance bounds. In this way, our results broaden the current understanding of double circulant and double negacirculant codes over local rings and provide new algebraically structured families of self-dual and LCD codes with good asymptotic performance.

This paper is structured as follows. Section 2 presents the notations and foundational settings pertinent to the subsequent sections of the paper. Section 3 examines the construction of double circulant and double negacirculant codes over $R_{q,u,v}$ and provides enumerations of such codes of length n in these settings. Section 4 presents distance bounds and shows that the Gray images of self-dual and LCD double circulant codes over $R_{q,u,v}$ form good code families. This section also includes several nontrivial examples over ${\mathbb{F}}_7$ obtained from the Gray images of such codes over $R_{q,u,v}$.

2. Preliminaries

Throughout, p is an odd prime number and q is taken such that the finite field ${\mathbb{F}}_q$ contains an element $\nu$ with ${\nu }^2=-1.$ In this case, the possible conditions for q are [19]

$$\left\{\begin{array}{cc}q=p^m,\hfill & p\equiv 1\left(\mathrm{mod}4\right);\hfill \\ q=p^{2m},\hfill & p\equiv 3\left(\mathrm{mod}4\right),m\ge 1.\hfill \end{array}\right.$$

(2)

In other words, we assume that q satisfies exactly one condition from above. This assumption will be used when solving equations such as $1+x^2=0.$ In this paper, we work over the commutative local ring $R_{q,u,v}={\displaystyle \frac{{\mathbb{F}}_q[u,v]}{\langle u^2,v^2,uv,vu\rangle }},$ i.e.,

$$R_{q,u,v}={\mathbb{F}}_q+u{\mathbb{F}}_q+v{\mathbb{F}}_q,$$

(3)

where u and v are satisfying $u^2=v^2=0=uv=vu$. Every element x of $R_{q,u,v}$ has a unique representation

$$x=a+ub+vc,$$

(4)

where $a,b,c\in {\mathbb{F}}_q.$ The ring $R_{q,u,v}$ is a local ring with maximal ideal $\mathfrak{m}=\langle u,v\rangle$ and the unit group is of the form

$$R_{q,u,v}^{\times }=\{a+ub+vc\mid a\in {\mathbb{F}}_q^{\times },b,c\in {\mathbb{F}}_q\},$$

with cardinality $\#\left\{R_{q,u,v}^{\times }\right\}=(q-1)q^2$.

We employ two ${\mathbb{F}}_q$-linear Gray maps from $R_{q,u,v}$ to ${\mathbb{F}}_q^3$:

$${\psi }_1(a+ub+vc)=(-b,2a+b,c),{\psi }_2(a+ub+vc)=(a,a+b,a+c)$$

(5)

both of which extend naturally to $R_{q,u,v}^n$ as ${\mathbb{F}}_q$-linear isomorphisms.

A linear code $\mathcal{C}\subseteq R_{q,u,v}^n$ is an $R_{q,u,v}$-submodule. For a vector $\mathbf{c}=(c_0,c_1,\dots ,c_{n-1})\in \mathcal{C}$, the Hamming weight $w_H\left(\mathbf{c}\right)$ is the number of non-zero coordinates, i.e., the number of i with $c_i\ne 0$. The Lee weight is

$$w_L\left(\mathbf{c}\right)=w_H\left({\psi }_i\left(\mathbf{c}\right)\right),i\in \{1,2\}.$$

(6)

The minimum Hamming and Lee distances, $d_H\left(\mathcal{C}\right)$ and $d_L\left(\mathcal{C}\right)$, are defined as the minimal non-zero Hamming and Lee weights in $\mathcal{C}$, respectively.

The Euclidean dual is

$${\mathcal{C}}^{\perp }=\{x\in R_{q,u,v}^n\mid \langle x,y\rangle =0\mathrm{for}\mathrm{all}y\in \mathcal{C}\},$$

(7)

where $\langle x,y\rangle$ is the standard dot product. The Hermitian dual ${\mathcal{C}}^{\perp H}$ uses the conjugate-linear inner product

$${\langle a+ub+vc,a_1+u{b}_1+v{c}_1\rangle }_H=a{a}_1^{\#}+ub{b}_1^{\#}+vc{c}_1^{\#},a^{\#}=a^{\sqrt{q}}.$$

(8)

A code is Euclidean (respectively, Hermitian) self-dual if it equals its Euclidean (respectively, Hermitian) dual and Euclidean (respectively, Hermitian) LCD if its intersection with the corresponding dual is trivial. It is known (see [Theorem 2.2] in [13]) that $\mathcal{C}$ is Euclidean LCD (respectively, self-dual) over $R_{q,u,v}$ if and only if both ${\psi }_1\left(\mathcal{C}\right)$ and ${\psi }_2\left(\mathcal{C}\right)$ are Euclidean LCD (respectively, self-dual) over ${\mathbb{F}}_q$, where the usual Euclidean inner product on ${\mathbb{F}}_q$ is used.

Finally, double circulant and double negacirculant codes over $R_{q,u,v}$ are defined via generator matrices

$$G=[I\mid M],$$

(9)

where M is a circulant or negacirculant matrix, respectively. An $n\times n$ matrix G is circulant if it has the form

$$G=\left(\begin{array}{cccc}{g}_0& g_1& \dots & g_{n-1}\\ g_{n-1}& g_0& \dots & g_{n-2}\\ ⋮& ⋮& \ddots & ⋮\\ g_1& g_2& \dots & g_0\end{array}\right),$$

(10)

where each subsequent row is obtained by a cyclic right shift of the previous row. An $n\times n$ matrix G is negacirculant if it has the form

$$G=\left(\begin{array}{ccccc}{g}_0& g_1& g_2& \dots & g_{n-1}\\ -g_{n-1}& g_0& g_1& \dots & g_{n-2}\\ -g_{n-2}& -g_{n-1}& g_0& \dots & g_{n-3}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ -g_1& -g_2& -g_3& \dots & g_0\end{array}\right),$$

(11)

where each subsequent row is obtained by a negacyclic shift (right shift with sign alternation) of the previous row. Double circulant and negacirculant codes are quasi-cyclic of index $2$.

3. Double Circulant and Double Negacirculant Codes

We first determine the number of solutions to significant equations over ${\mathbb{F}}_q$, which will be utilized later in the enumeration sections.

Lemma 1.

Let q be an odd prime power. The equation $a{b}^q+b{a}^q=0$ in ${\mathbb{F}}_{q^2}$ has exactly $(q^2-1)q$ solutions with $a\ne 0$.

Proof. 

We now analyze the equation $a{b}^q+b{a}^q=0$ for $a\ne 0$ in detail. First, since $a\ne 0$, we can divide both sides of the equation by $a^{q+1}$ to obtain $\frac{b^q}{a^q}+\frac{b}{a}=0$. Let $c=\frac{b}{a}$. Then, the equation simplifies to $c^q+c=0$. We study solutions to $c^q=-c$. The trivial solution is $c=0$. For $c\ne 0$, we can write the equation as $c^{q-1}=-1$. To count the number of non-zero solutions, we use the following observations about the field ${\mathbb{F}}_{q^2}$. The multiplicative group ${\mathbb{F}}_{q^2}^{\times }$ is cyclic of order $q^2-1$. Moreover, the equation $c^{q-1}=-1$ has solutions if and only if $-1$ is in the image of the $(q-1)$-th power map. Suppose that $c={\omega }^k$, where $\langle \omega \rangle ={\mathbb{F}}_{q^2}^{\times }$. Then, $c^{q-1}={\omega }^{k(q-1)}=-1$, and thus $k(q-1)=\frac{q^2-1}{2}mod{q}^2-1$. The solution exists if and only if $gcd(q-1,q^2-1)$ divides $\frac{q^2-1}{2}$. As $gcd(q-1,q^2-1)=q-1$ divides ${\displaystyle \frac{q^2-1}{2}}$, we have the existence of the solution of $c^{q-1}=-1.$ The number of solutions to $c^{q-1}=-1$ is equal to the number of $(q-1)$-th roots of $-1$. Since the $(q-1)$-th power map is a group homomorphism ${\mathbb{F}}_{q^2}^{\times }\to {\mathbb{F}}_{q^2}^{\times }$ with a kernel of size $gcd(q-1,q^2-1)=q-1$, the equation $c^{q-1}=-1$ has exactly $q-1$ solutions when $-1$ is in the image, which it is, as established. Therefore, the equation $c^q=-c$ has 1 solution when $c=0$ and $q-1$ solutions when $c\ne 0$, giving a total of q solutions for c. The solutions are ${\omega }^{k+i\frac{q^2-1}{q-1}}$, where $i=0,1,2,\dots ,q-1.$

For each non-zero $a\in {\mathbb{F}}_{q^2}$, there are exactly q corresponding b values (one for each solution c, via $b=ca$). Since there are $q^2-1$ choices for $a\ne 0$, the total number of solutions $(a,b)$ with $a\ne 0$ is $(q^2-1)q$. □

Remark 1.

The equation is $c^q+c=0$ over ${\mathbb{F}}_{q^2}$, where $q=p^m$ and $p\ne 2$ defines a 1-dimensional ${\mathbb{F}}_q$-subspace of ${\mathbb{F}}_{q^2}$. Indeed, viewing the ${\mathbb{F}}_q$-linear Frobenius operator $\sigma :c\mapsto c^q$, its minimal polynomial over ${\mathbb{F}}_q$ is $z^2-1$, which factors into distinct roots since $p\ne 2$. Consequently, the endomorphism $\sigma +\mathrm{id}$ has a 1-dimensional kernel, yielding exactly q solutions to $c^q+c=0$.

This algebraic structure has a geometric interpretation in the affine space ${\mathbb{A}}^2\left({\mathbb{F}}_{q^2}\right)$ via the equation $a{b}^q+b{a}^q=0$. For a fixed $a\ne 0$, all solutions b can be written as $b=ay$ with y in the kernel above. Thus, the set of solutions forms an ${\mathbb{F}}_q$-line, reflecting the principles of Hermitian duality in ${\mathbb{F}}_{q^2}$. Moreover, the equation $a{b}^q+b{a}^q=0$ can be regarded as the alternating form associated with the Hermitian pairing $a{b}^q-b{a}^q$, whose solutions are precisely the isotropic vectors, i.e., self-orthogonal vectors with respect to this pairing.

This perspective unifies three viewpoints: the arithmetic of ${\mathbb{F}}_{q^2}$ under the Frobenius action, the geometry of ${\mathbb{F}}_q$-lines in affine space, and the algebra of quadratic forms. The recurrent appearance of expressions of the form $a{b}^q\pm b{a}^q$ illustrates the deep interplay between these areas, as linear algebra over ${\mathbb{F}}_q$ constrains both the dimension of the solution space and its Hermitian interpretation.

In what follows, we need the norm function $N:{\mathbb{F}}_{q^e}\to {\mathbb{F}}_q$, given by $N\left(\delta \right)={\delta }^{\frac{q^e-1}{q-1}}$, which is multiplicative and surjective, with each $\delta \in {\mathbb{F}}_q^{\times }$ having $\frac{q^e-1}{q-1}$ preimages in ${\mathbb{F}}_{q^e}^{\times }$.

Proposition 1.

The number of solutions in ${\mathbb{F}}_{q^2}$ of the system $1+a{a}^q=0,a{b}^q+b{a}^q=0,a{c}^q+c{a}^q=0$ is $(q+1)q^2.$

Proof. 

The system of equations reduces to two fundamental conditions: the norm condition $\mathrm{Norm}\left(a\right)=a{a}^q=-1$ and the trace-like conditions $\mathrm{Tr}\left(a{b}^q\right)=\mathrm{Tr}\left(a{c}^q\right)=0$. The norm equation $\mathrm{Norm}\left(a\right)=-1$ admits exactly $q+1$ solutions in ${\mathbb{F}}_{q^2}$, since the norm map $N:{\mathbb{F}}_{q^2}\to {\mathbb{F}}_q$ is surjective and each non-zero value has $q+1$ preimages. For each fixed solution a satisfying the norm condition, the constraints on b and c become ${\mathbb{F}}_q$-linear equations. These determine affine subspaces of dimension 1 over ${\mathbb{F}}_q$ (1-dimensional ${\mathbb{F}}_q$-subspace of ${\mathbb{F}}_{q^2}$), resulting in q valid choices for each of b and c independently by Lemma 1 and Remark 1. Consequently, the total number of admissible triples $(a,b,c)$ is given by the product

$$(q+1)\times q\times q=(q+1)q^2.$$

□

3.1. Structures and Enumeration of Double Circulant Codes

Let n be an odd positive integer, $gcd(n,q)=1$, and over $R_{q,u,v}$, we have

$$z^n-1=\delta (z-1)\prod _{i=2}^e{l}_i\left(z\right)\prod _{j=1}^d{k}_j\left(z\right)k_j^{\ast }\left(z\right),$$

(12)

where

$$\left\{\begin{array}{c}n=1+\sum _{i=2}^{e}2f_i+2\sum _{j=1}^z{h}_j,\hfill \\ \delta \in R_{q,u,v}^{\times }(\mathrm{the}\mathrm{group}\mathrm{of}\mathrm{units}\mathrm{in}{R}_{q,u,v});\hfill \\ k_j^{\ast }\left(z\right)\mathrm{is}\mathrm{the}\mathrm{reciprocal}\mathrm{of}{k}_j\left(z\right),\mathrm{both}\mathrm{of}\mathrm{degree}{h}_j\mathrm{for}1\le j\le d;\hfill \\ l_i\left(z\right)\mathrm{is}\mathrm{self}\text{-}\mathrm{reciprocal}\mathrm{of}\mathrm{degree}2f_i\mathrm{for}2\le i\le e.\hfill \end{array}\right.$$

By CRT, we obtain the decomposition

$${\displaystyle \frac{R_{q,u,v}\left[z\right]}{\langle z^n-1\rangle }}\cong R_{q,u,v}\oplus \left(\underset{i=2}{\overset{e}{⨁}}{\displaystyle \frac{R_{q,u,v}\left[z\right]}{\langle l_i\left(z\right)\rangle }}\right)\oplus \left(\underset{j=1}{\overset{d}{⨁}}\left({\displaystyle \frac{R_{q,u,v}\left[z\right]}{\langle k_j\left(z\right)\rangle }}\oplus {\displaystyle \frac{R_{q,u,v}\left[z\right]}{\langle k_j^{\ast }\left(z\right)\rangle }}\right)\right)$$

(13)

with $R_{q^r,u,v}={\mathbb{F}}_{q^r}+u{\mathbb{F}}_{q^r}+v{\mathbb{F}}_{q^r}$ for $r\in \{2f_i,h_j\}$, where $u^2=u$, $v^2=0$, and $uv=vu=0$. This extends to

$${\left({\displaystyle \frac{R_{q,u,v}\left[z\right]}{\langle z^n-1\rangle }}\right)}^2\cong R_{q,u,v}^2\oplus \left(\underset{i=2}{\overset{e}{⨁}}{R}_{q^{2f_i},u,v}^2\right)\oplus \left(\underset{j=1}{\overset{d}{⨁}}\left(R_{q^{h_j},u,v}^2\oplus R_{q^{h_j},u,v}^2\right)\right).$$

(14)

Any length-2 linear code $\mathcal{C}$ over $\frac{R_{q,u,v}\left[z\right]}{\langle z^n-1\rangle }$ decomposes as

$$\mathcal{C}\cong {\mathcal{C}}_1\oplus \left(\underset{i=2}{\overset{e}{⨁}}{\mathcal{C}}_i\right)\oplus \left(\underset{j=1}{\overset{d}{⨁}}\left({\mathcal{C}}_j\oplus {\mathcal{C}}_j^{\perp }\right)\right),$$

(15)

$$\left\{\begin{array}{c}{\mathcal{C}}_1\mathrm{is}\mathrm{a}\mathrm{code}\mathrm{over}{R}_{q,u,v},\hfill \\ {\mathcal{C}}_i\mathrm{is}\mathrm{a}\mathrm{code}\mathrm{over}{R}_{2f_i}\mathrm{for}2\le i\le e,\hfill \\ {\mathcal{C}}_j,{\mathcal{C}}_j^{\perp }\mathrm{are}\mathrm{codes}\mathrm{over}{R}_{h_j}\mathrm{for}1\le j\le d.\hfill \end{array}\right.$$

Proposition 2.

Let $\mathcal{C}$ be a double circulant code over $R_{q,u,v}$ decomposed as in (), and

$$\begin{array}{cc}\hfill {\mathcal{C}}_1& =\langle (1,c_{f_1})\rangle \subset R_{q,u,v},{\mathcal{C}}_i=\langle (1,c_{f_i})\rangle \subset R_{q^{2f_i},u,v}\mathrm{for}2\le i\le e,\hfill \\ \hfill {\mathcal{C}}_j^{\left(1\right)}& =\langle (1,c_{h_j}^{\left(1\right)})\rangle \subset R_{q^{h_j},u,v},{\mathcal{C}}_j^{\left(2\right)}=\langle (1,c_{h_j}^{\left(2\right)})\rangle \subset R_{q^{h_j},u,v}\mathrm{for}1\le j\le d.\hfill \end{array}$$

Then, the following characterizations hold:

1. 

The code $\mathcal{C}$ is self-dual if and only if

$$\begin{array}{cc}\hfill & 1+c_{f_1}^2=0,\mathit{in}{R}_{q,u,v};\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & 1+c_{f_i}^{1+q^{f_i}}=0,\mathit{in}{R}_{q^{2f_i},u,v}(2\le i\le e);\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & 1+c_{h_j}^{\left(1\right)}{c}_{h_j}^{\left(2\right)}=0,\mathit{in}{R}_{q^{h_j},u,v}(1\le j\le d).\hfill \end{array}$$

(18)

2. 

The code $\mathcal{C}$ is Euclidean LCD if and only if the following units exist:

$$\begin{array}{c}\hfill 1+c_{f_1}^2\in R^{\times },1+c_{f_i}^{1+q^{f_i}}\in R_{q^{2f_i},u,v}^{\times },1+c_{h_j}^{\left(1\right)}{c}_{h_j}^{\left(2\right)}\in R_{q^{h_j},u,v}^{\times }.\end{array}$$

(19)

Proof. 

The result follows from the CRT decomposition in (13) and the characterization of self-duality and the LCD property for double circulant codes. For a double circulant code $\mathcal{C}$ with generator matrix $G=[I\mid M]$, the self-duality condition $\mathcal{C}={\mathcal{C}}^{\perp }$ is equivalent to $G{G}^{\top }=0$. Direct computation gives $G{G}^{\top }=[I\mid M]{[I\mid M]}^{\top }=I+M{M}^{\top }$. Thus, self-duality requires $M{M}^{\top }=-I$. Since M is circulant and corresponds to an element c over $R_{q,u,v}$, this matrix equation translates to the polynomial equation $1+c^2=0$ in the ring. Under the CRT decomposition, this condition decouples into independent conditions on each component:

• For the component over $R_{q,u,v}$ corresponding to the linear factor, self-duality requires $1+c_{f_1}^2=0$;
• For components over $R_{q^{2f_i},u,v}$ corresponding to self-reciprocal irreducible factors, self-duality requires $1+c_{f_i}^{1+q^{f_i}}=0$, where the exponent $1+q^{f_i}$ arises from the Frobenius action;
• For reciprocal pairs over $R_{q^{h_j},u,v}$, self-duality requires $1+c_{h_j}^{\left(1\right)}{c}_{h_j}^{\left(2\right)}=0$.

Similarly, $\mathcal{C}$ is Euclidean LCD if and only if $1+c^2$ is a unit in the appropriate ring. Under the CRT decomposition, this becomes the requirement that each component $1+c_{f_1}^2$, $1+c_{f_i}^{1+q^{f_i}}$, and $1+c_{h_j}^{\left(1\right)}{c}_{h_j}^{\left(2\right)}$ is a unit in its respective ring. The result follows by applying these characterizations to each component of the CRT decomposition. □

Theorem 1.

Suppose that n is an odd integer, and consider factorization (12). Then, the total number ${\mathcal{N}}_s$ of self-dual double circulant codes over $R_{q,u,v}$ is

$${\mathcal{N}}_s=2\prod _{i=2}^e[q^{3f_i}+q^{2f_i}]\prod _{j=1}^d\left[(q^{h_j}-1)q^{2h_j}\right].$$

(20)

Proof. 

The value of ${\mathcal{N}}_s$ can be determined through a counting method. There are only two options for ${\mathcal{C}}_1$, characterized by the generator polynomials $(1,\nu )$ and $(1,-\nu )$, where ${\nu }^2=-1$. To determine the count of self-dual codes concerning the Hermitian inner product, it is necessary to identify the count of zeros of

$$1+c_{f_i}{c}_{f_i}^{q^{f_i}}=0.$$

(21)

Expressing $c_{f_i}$ in terms of the basis elements, we write

$$c_{f_i}=a+bu+cv\mathrm{for}\mathrm{some}a,b,c\in {\mathbb{F}}_{q^{2f_i}}.$$

Substituting this expression into our original equation yields

$$1+\left(a+ub+vc\right){\left(a+ub+vc\right)}^{q^{f_i}}=0.$$

Let us analyze in detail the number of solutions to the equation $1+(a+bu+cv){(a+bu+cv)}^{q^{f_i}}=0$ in the ring $R_{q,u,v}$, where $a,b,c\in {\mathbb{F}}_{q^{2f_i}}$ and $u^2=0$. First, we expand the product using the ring relations

$$\begin{array}{cc}\hfill (a+bu+cv)(a^{q^{f_i}}+b^{q^{f_i}}u+c^{q^{f_i}}v)& =a{a}^{q^{f_i}}+a{b}^{q^{f_i}}u+b{a}^{q^{f_i}}u+a{c}^{q^{f_i}}v+c{a}^{q^{f_i}}v\hfill \\ \hfill & =a{a}^{q^{f_i}}+(a{b}^{q^{f_i}}+b{a}^{q^{f_i}})u+(a{c}^{q^{f_i}}+c{a}^{q^{f_i}})v\hfill \end{array}$$

since all higher-order terms vanish ($u^2=0$, $v^2=0$, $uv=0$). This gives us the equation

$$1+a{a}^{q^{f_i}}+(a{b}^{q^{f_i}}+b{a}^{q^{f_i}})u+(a{c}^{q^{f_i}}+c{a}^{q^{f_i}})v=0.$$

Equating the coefficients yields three independent conditions.

• The constant term: It gives the norm equation $1+a{a}^{q^{f_i}}=0.$ This is the norm condition $N\left(a\right)=-1$, where $N:{\mathbb{F}}_{q^{2f_i}}\to {\mathbb{F}}_{q^{f_i}}$ is the field norm. Since the norm is surjective and each value has $q^{f_i}+1$ preimages, there are exactly $q^{f_i}+1$ solutions for $a$.
• The u-coefficient gives

  $$a{b}^{q^{f_i}}+b{a}^{q^{f_i}}=0.$$
  For each fixed $a\ne 0$, it has exactly $q^{f_i}$ solutions for b by Lemma 1.
• The v-coefficient gives

  $$a{c}^{q^{f_i}}+c{a}^{q^{f_i}}=0.$$
  This has an identical structure to the u-condition, providing $q^{f_i}$ solutions for c for each valid $(a,b)$ pair.

Thus, the total number of solutions $(a,b,c)$ is

$$(q^{f_i}+1)\times q^{f_i}\times q^{f_i}=(q^{f_i}+1)q^{2f_i}.$$

We now enumerate the dual pairs of codes with respect to the Euclidean inner product. This requires determining the number of solutions of

$$1+c_{h_j}^{\left(1\right)}{c}_{h_j}^{\left(2\right)}=0.$$

(22)

Two disjoint cases arise depending on whether $c_{h_j}^{\left(1\right)}$ is a unit.

• Unit case: When $c_{h_j}^{\left(1\right)}\in R_{q,u,v,h_j}^{\times }$, the dual element must satisfy $c_{h_j}^{\left(2\right)}=-\frac{1}{c_{h_j}^{\left(1\right)}}$. In this scenario, there are exactly $(q^{h_j}-1)q^{2h_j}$ possible pairs $\{c_{h_j}^{\left(1\right)},c_{h_j}^{\left(2\right)}\}$, corresponding to the order of the unit group $R_{q,u,v,h_j}^{\times }$.
• Non-unit case: For $c_{h_j}^{\left(1\right)}\in R_{q,u,v,h_j}\setminus R_{q,u,v,h_j}^{\times }$, we express the element as

  $$c_{h_j}^{\left(1\right)}=bu+cv\mathrm{with}b,c\in {\mathbb{F}}_{q^{h_j}},$$

  where at least one coordinate is zero. Letting $c_{h_j}^{\left(2\right)}=b_1+b_{2}u+b_{3}v$, where $b_i\in {\mathbb{F}}_{q^{h_j}}$, the equation becomes

  $$1+c_{h_j}^{\left(1\right)}{c}_{h_j}^{\left(2\right)}=1+\left(b{b}_1\right)u+\left(c{b}_1\right)v=0.$$
  This yields the system

  $$b{b}_1=0,c{b}_1=0.$$

  (23)
  However, we also obtain the contradiction $1=0$. Thus, no solutions exist in this case.

The complete enumeration shows that only the unit case contributes valid solutions, giving exactly $(q^{h_j}-1)q^{2h_j}$ distinct dual pairs. Combining these results yields the desired conclusion. □

Theorem 2.

The number of LCD double circulant codes over $R_{q,u,v}$ is given by

$${\mathcal{N}}_d=(q-2)q^2\prod _{i=2}^e\left[(q^{2f_i}-q^{f_i}-1)q^{4f_i}\right]\prod _{j=1}^z[q^{3h_j}+{\left((q^{h_j}-1)q^{h_j}\right)}^2].$$

(24)

Proof. 

If ${\mathcal{C}}_1$ is an LCD code, first, we recall that q is odd and $-1$ is a square in ${\mathbb{F}}_q$. We determine the number of triples $(a,b,c)\in {\mathbb{F}}_q^3$ for which $1+{(a+bu+cv)}^2$ is a unit in R. Expanding ${(a+bu+cv)}^2$ yields $a^2+2abu+2acv$ since all higher-order terms vanish due to the ring relations. The element $1+{(a+bu+cv)}^2=(1+a^2)+2abu+2acv$ is a unit if and only if its constant term $1+a^2\ne 0$. When $-1$ is a square, there exist exactly two distinct elements $\nu ,-\nu \in {\mathbb{F}}_q$ such that ${\nu }^2=-1$. Thus, $1+a^2=0$ holds only when $a=\nu$ or $a=-\nu$, leaving $q-2$ valid choices for a. For each such a, the coefficients b and c can be chosen freely from ${\mathbb{F}}_q$ because the nilpotent terms $2abu$ and $2acv$ do not affect the invertibility condition. Consequently, there are q options for b and q for c per valid a, resulting in $(q-2)\times q\times q=(q-2)q^2$ total solutions.

Under the Hermitian inner product, we determine when ${\mathcal{C}}_{f_i}$ is an LCD code over $R_{q^{2f_i},u,v}$. Thus, ${\mathcal{C}}_{f_i}$ is Hermitian LCD if $1+{(a+bu+cv)}^{1+q^{f_i}}$ is a unit in $R_{q^{2f_i},u,v}$. Expanding using the ring relations, we compute

$$\begin{array}{cc}\hfill {(a+bu+cv)}^{1+q^{f_i}}& =(a+bu+cv){(a+bu+cv)}^{q^{f_i}}\hfill \\ \hfill & =(a+bu+cv)(a^{q^{f_i}}+b^{q^{f_i}}u+c^{q^{f_i}}v)\hfill \\ \hfill & =a{a}^{q^{f_i}}+(a{b}^{q^{f_i}}+b{a}^{q^{f_i}})u+(a{c}^{q^{f_i}}+c{a}^{q^{f_i}})v\hfill \end{array}$$

Thus, the expression becomes

$$1+{(a+bu+cv)}^{1+q^{f_i}}=(1+a{a}^{q^{f_i}})+(a{b}^{q^{f_i}}+b{a}^{q^{f_i}})u+(a{c}^{q^{f_i}}+c{a}^{q^{f_i}})v$$

For this to be a unit in R, the constant term must be non-zero, i.e., $1+a{a}^{q^{f_i}}\ne 0$. This is equivalent to requiring that a is not a root of the equation $a{a}^{q^{f_i}}=-1$. The number of such $a\in {\mathbb{F}}_{q^{2f_i}}$ is exactly $q^{f_i}+1$. Therefore, there are $q^{2f_i}-(q^{f_i}+1)=q^{2f_i}-q^{f_i}-1$ valid a values. For each valid a, the coefficients b and c can be chosen freely from ${\mathbb{F}}_{q^{2f_i}}$, giving $q^{2f_i}$ choices for each using Proposition 1. Therefore, the total number of triples $(a,b,c)$ is

$$(q^{2f_i}-q^{f_i}-1)\times q^{2f_i}\times q^{2f_i}=(q^{2f_i}-q^{f_i}-1)q^{4f_i}.$$

We analyze the condition for $1+(a+bu+cv)(a^{\prime }+b^{\prime }u+c^{\prime }v)$ to be a unit in the ring $R_{q^{h_j},u,v}$. Expanding the product gives $(a+bu+cv)(a^{\prime }+b^{\prime }u+c^{\prime }v)=a{a}^{\prime }+(a{b}^{\prime }+b{a}^{\prime })u+(a{c}^{\prime }+c{a}^{\prime })v$ since all higher-order terms vanish due to the ring relations. The expression then becomes $1+(a+bu+cv)(a^{\prime }+b^{\prime }u+c^{\prime }v)=(1+a{a}^{\prime })+(a{b}^{\prime }+b{a}^{\prime })u+(a{c}^{\prime }+c{a}^{\prime })v$. For this to be a unit in R, the constant term must satisfy $1+a{a}^{\prime }\ne 0$ in ${\mathbb{F}}_{q^{h_j}}$, while the coefficients of u and v do not affect invertibility as they multiply nilpotent elements. When $a=0$, the condition reduces to $c_{f_i}^{\left(2\right)}$, which means that there are $q^{3h_j}$ of $a^{\prime },b^{\prime }$ and $c^{\prime }.$ For $a\ne 0$, we must have $a^{\prime }\ne -a^{-1},0$, yielding $q^{h_j}-2$ valid choices for $a^{\prime }$ for each of the $q^{h_j}-1$ non-zero a. The coefficients $b,b^{\prime },c,c^{\prime }$ are independent and freely chosen from ${\mathbb{F}}_{q^{h_j}}$, with $q^{h_j}$ choices each. The total number of solutions is therefore $(q^{h_j}-2)(q^{h_j}-1)q^{4h_j}$, computed as $(q^{h_j}-1)$ choices for a, $(q^{h_j}-1)$ for $a^{\prime }$, and $q^{h_j}$ for each of $b,b^{\prime },c,c^{\prime }$. This count holds uniformly across all finite fields ${\mathbb{F}}_{q^{h_j}}$ regardless of the characteristic. The final result is

$$q^{3h_j}+{((q^{h_j}-1)q^{h_j})}^2.$$

The complete count is obtained by multiplying together the enumerations from all three cases: the $(q-2)q^2$ Euclidean LCD components, the product of Hermitian LCD components ${\prod }_{i=2}^e\left[(q^{2f_i}-q^{f_i}-1)q^{4f_i}\right]$, and the product of dual pair components $(q^{h_j}-2)(q^{h_j}-1)q^{4h_j}$. This implies

$${\mathcal{N}}_d=\underset{\mathrm{Case}1}{\underbrace{\left[(q-2)q^2\right]}}\times \underset{\mathrm{Case}2}{\underbrace{\prod _{i=2}^e\left[(q^{2f_i}-q^{f_i}-1)q^{4f_i}\right]}}\times \underset{\mathrm{Case}3}{\underbrace{\prod _{j=1}^z[q^{3h_j}+{\left((q^{h_j}-1)q^{h_j}\right)}^2]}}$$

where each factor corresponds to a distinct class of components in the decomposition. Combining the independent component counts yields the stated formula. The product structure reflects the independent choices available for each type of component while maintaining the necessary algebraic constraints. □

Example 1.

The factorization of $z^{15}-1$ over ${\mathbb{F}}_7$ is given by

$$\begin{array}{c}{z}^{15}-1=(z+6)(z+3)(z+5)(z^4+z^3+z^2+z+1)\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(z^4+2z^3+4z^2+z+2)(z^4+4z^3+2z^2+z+4).\end{array}$$

(25)

The factors decompose as follows: the linear term $z+6=z-1\left(\mathrm{mod}7\right)$ serves as the special term; $(z+3,z+5)$ form a reciprocal pair since $3\times 5=1\left(\mathrm{mod}7\right)$; $p\left(z\right)=z^4+z^3+z^2+z+1$ is self-reciprocal with $z^{4}p\left(\frac{1}{z}\right)=p\left(z\right)$; and $(k\left(z\right)=z^4+2z^3+4z^2+z+2,k^{\ast }\left(z\right)=z^4+4z^3+2z^2+z+4)$ form another reciprocal pair after coefficient normalization. The degree formula is verified as

$$n=1+\sum _{i=1}^{e}2f_i+2\sum _{j=1}^z{h}_j=1+2\left(2\right)+2(1+4)=15,$$

where $f_1=2$ corresponds to a self-reciprocal quartic factor, $h_1=1$ to a linear conjugate pair, and $h_2=4$ to a quartic conjugate pair. The unit count is then obtained by applying Theorem 1,

$${\mathcal{N}}_s=2(7^2+7)(7-1)\left(7^2\right)(7^4-1)\left(7^8\right)=96\left(7^{11}\right)(7^4-1).$$

Theorem 2 for constrained invertibles gives

$${\mathcal{N}}_d=215\left[7^{11}·(7^4-7^2-1\right]·\left[7^{12}+{(7^4-1)}^2·7^4\right].$$

This demonstrates how the factorization structure determines both counting formulas, with the special term $z+6=z-1\left(\mathrm{mod}7\right)$ ensuring that the degree formula closes correctly.

3.2. Structures and Enumeration of Double Negacirculant Codes

This subsection focuses on enumerating self-dual and LCD double negacirculant codes over the ring $R_{q,u,v}$. We consider even positive integers n satisfying $gcd(n,q)=1$, and, over $R_{q,u,v},$ suppose

$$z^n+1=\gamma \prod _{i=1}^e{l}_i\left(z\right)\prod _{j=1}^d{k}_j\left(z\right)k_j^{\ast }\left(z\right),$$

(26)

where the factors satisfy

$$\left\{\begin{array}{cc}\gamma \in R_{q,u,v}^{\ast },\hfill & (\mathrm{unit}\mathrm{constant})\hfill \\ l_i\left(z\right)=z^{2f_i}{l}_i\left(\frac{1}{z}\right),deg\left(l_i\right)=2f_i\hfill & \left(\mathrm{self}\text{-}\mathrm{reciprocal}\right)\hfill \\ k_j^{\ast }\left(z\right)=z^{h_j}{k}_j\left(\frac{1}{z}\right),deg\left(k_j\right)=deg\left(k_j^{\ast }\right)=h_j\hfill & (\mathrm{reciprocal}\mathrm{pair})\hfill \end{array}\right.$$

Following a methodology analogous to that in the double circulant case, we obtain the ring decomposition

$${\displaystyle \frac{R_{q,u,v}\left[z\right]}{\langle z^n+1\rangle }}\cong \underset{i=1}{\overset{e}{⨁}}{R}_{q,u,v}^{2f_i}\oplus \underset{j=1}{\overset{d}{⨁}}(R_{q,u,v}^{h_j}\oplus R_{q,u,v}^{h_j}),$$

where for $r=2f_i$ or $h_j$, and the components $R_{q^r,u,v}={\mathbb{F}}_{q^r}+u{\mathbb{F}}_{q^r}+v{\mathbb{F}}_{q^r}$ satisfy the relations $u^2=0$, $v^2=0$, and $uv=vu=0$. Consequently, any length-2 linear code C decomposes as

$$C\cong \underset{i=1}{\overset{e}{⨁}}{\mathcal{C}}_i\oplus \underset{j=1}{\overset{d}{⨁}}({\mathcal{C}}_j^{\left(1\right)}\oplus {\mathcal{C}}_j^{\left(2\right)}),$$

with ${\mathcal{C}}_i$ being a linear code over $R_{q,u,v}^{2f_i}$ for $1\le i\le e$ and ${\mathcal{C}}_j^{\left(1\right)},{\mathcal{C}}_j^{\left(2\right)}$ linear codes over $R_{q,u,v}^{h_j}$ for $1\le j\le d$.

The enumeration is built upon the following key result, which follows from arguments similar to those involved in establishing Proposition 2.

Lemma 2.

Let C be a double negacirculant code over $R_{q,u,v}$ with code codes ${\mathcal{C}}_i\subseteq R_{q^{2f_i},u,v}$ ($1\le i\le e$) and ${\mathcal{C}}_j^{\left(1\right)},{\mathcal{C}}_j^{\left(2\right)}\subseteq R_{q^{h_j},u,v}$ ($1\le j\le d$). Suppose that the generators of these codes are given by

$$\begin{array}{cc}\hfill c_i& =(1,c_{f_i}),\mathrm{for}{\mathcal{C}}_i,\hfill \\ \hfill c_j^{\left(1\right)}& =(1,c_{h_j}^{\left(1\right)}),\mathrm{for}{\mathcal{C}}_j^{\left(1\right)},\hfill \\ \hfill c_j^{\left(2\right)}& =(1,c_{h_j}^{\left(2\right)}),\mathrm{for}{\mathcal{C}}_j^{\left(2\right)}.\hfill \end{array}$$

Then, the following characterizations hold.

1. 

Self-duality: The code C is self-dual if and only if the following conditions are satisfied:

$$\begin{array}{cc}\hfill 1+c_{f_i}^{1+q^{f_i}}& =0,\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill 1+c_{h_j}^{\left(1\right)}{c}_{h_j}^{\left(2\right)}& =0.\hfill \end{array}$$

(28)

2. 

Euclidean LCD property: The code C is Euclidean LCD if and only if

$$\begin{array}{cc}\hfill 1+c_{f_i}^{1+q^{f_i}}& \in R_{q^{2f_i},u,v}^{\times },\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill 1+c_{h_j}^{\left(1\right)}{c}_{h_j}^{\left(2\right)}& \in R_{q^{h_j},u,v}^{\times }.\hfill \end{array}$$

(30)

Theorem 3.

For an even integer n, suppose the factorization in (26). The number ${\mathcal{N}}_{snega}$ of self-dual double negacirculant codes over R is

$${\mathcal{N}}_{snega}=\prod _{i=1}^e(q^{f_i}+1)q^{2f_i}\prod _{j=1}^d(q^{h_j}-1)q^{2h_j}.$$

(31)

Proof. 

The enumeration of ${\mathcal{N}}_{snega}$ reduces to counting solutions for the code codes. For each code code ${\mathcal{C}}_i$, we must determine the number of solutions to the algebraic condition

$$1+c_{f_i}{c}_{f_i}^{q^{f_i}}=0,$$

where $c_{f_i}\in R^{2f_i}$. Representing $c_{f_i}$ in terms of the basis $\{1,u,v\}$, we write

$$c_{f_i}=a+bu+cv\mathrm{for}a,b,c\in {\mathbb{F}}_{q^{2f_i}}.$$

Substituting this expression into our equation yields

$$1+(a+bu+cv)(a^{q^{f_i}}+b^{q^{f_i}}u+c^{q^{f_i}}v)=0.$$

Expanding the product and using the ring relations $u^2=v^2=uv=0$, we obtain the system

$$\begin{array}{c}\hfill 1+a{a}^{q^{f_i}}=0,a{b}^{q^{f_i}}+b{a}^{q^{f_i}}=0,a{c}^{q^{f_i}}+c{a}^{q^{f_i}}=0.\end{array}$$

Consequently, by Proposition 1, the total number of admissible triples $(a,b,c)$ is given by the product

$$(q^{f_i}+1)\times q^{f_i}\times q^{f_i}=(q^{f_i}+1)q^{2f_i}.$$

To enumerate the dual pairs $\{c_j^{\left(1\right)},c_j^{\left(2\right)}\}$, we analyze solutions to the equation

$$1+c_{h_j}^{\left(1\right)}{c}_{h_j}^{\left(2\right)}=0.$$

The solution space decomposes into two cases based on the unit properties of $c_{h_j}^{\left(1\right)}$.

• Unit case: When $c_{h_j}^{\left(1\right)}\in R_{q^{h_i},u,v}^{\times }$, the relation $c_{h_j}^{\left(2\right)}=-\frac{1}{c_{h_j}^{\left(1\right)}}$ determines a unique dual partner. Since the unit group satisfies $|R_{q^{h_i},u,v}^{\times }|=(q^{h_j}-1)q^{2h_j}$, there are exactly $(q^{h_j}-1)q^{2h_j}$ valid pairs in this case.
• Non-unit case: For $c_{h_j}^{\left(1\right)}\in R_{q^{h_i},u,v}\setminus R_{q^{h_i},u,v}^{\times }$, expressed as $a+bu+cv$, where $a,b,c\in {\mathbb{F}}_{q^{h_j}}$ are not all zero, the equation becomes

  $$1+(a+bu+cv)(b_1+b_{2}u+b_{3}v)=(1+a{b}_1)+(b{b}_1+a{b}_2)u+(c{b}_1+a{b}_3)v=0.$$
  This system requires

  $$\begin{array}{c}\hfill 1+a{b}_1=0,b{b}_1+a{b}_2=0,c{b}_1+a{b}_3=0.\end{array}$$
  If any coefficient $a,b,c$ vanishes, the system becomes inconsistent ($1=0$). Hence, no solutions exist in the non-unit case.

Combining these results with the previous enumeration of ${\mathcal{C}}_i$ components, we have

$${\mathcal{N}}_{snega}=\prod _{i=1}^e(q^{f_i}+1)q^{2f_i}\prod _{j=1}^d(q^{h_j}-1)q^{2h_j}.$$

□

Using an analogous methodology, we now establish the enumeration of LCD double negacirculant codes over $R_{q,u,v}$.

Theorem 4.

The total number of LCD double negacirculant codes ${\mathcal{N}}_{dnega}$ over $R_{q,u,v}$ is

$${\mathcal{N}}_{dnega}=\prod _{i=1}^e(q^{6f_i}-q^{5f_i}-2q^{3f_i}-6q^{2f_i}+5q^{f_i}+8)\prod _{j=1}^d(q^{6h_j}-2q^{5h_j}+q^{4h_j}+q^{3h_j}).$$

(32)

Proof. 

A double negacirculant code over $R_{q,u,v}$ decomposes into a family of constituent codes

$$\left({\mathcal{C}}_1,\dots ,{\mathcal{C}}_e,{\{{\mathcal{C}}_j^{\left(1\right)},{\mathcal{C}}_j^{\left(2\right)}\}}_{j=1}^d\right),$$

where each ${\mathcal{C}}_i$ is a length-two code over $R_{q^{2f_i},u,v}$ and each pair $({\mathcal{C}}_j^{\left(1\right)},{\mathcal{C}}_j^{\left(2\right)})$ is defined over $R_{q^{h_j},u,v}$.

• (i) LCD condition for self-reciprocal constituents. For a constituent ${\mathcal{C}}_i=\langle (1,c_{f_i})\rangle$ over $R_{q^{2f_i},u,v}$, the LCD property holds precisely when

  $$1+c_{f_i}{c}_{f_i}^{\#}\in R_{q^{2f_i},u,v}^{\times },$$

  where $c_{f_i}^{\#}$ denotes the Hermitian conjugate with respect to the extension ${\mathbb{F}}_{q^{2f_i}}/{\mathbb{F}}_{q^{f_i}}$. Since $c_{f_i}^{\#}=c_{f_i}^{q^{f_i}}$, the condition is equivalently

  $$1+c_{f_i}^{1+q^{f_i}}\in R_{q^{2f_i},u,v}^{\times }.$$
• Case A: $c_{f_i}=0$. In this case, the condition is trivially satisfied because $1\in R^{\times }$. Hence, there is exactly one valid choice. Case B: $c_{f_i}\in {\mathbb{F}}_{q^{2f_i}}^{\times }$. Here, the constraint reads $a^{1+q^{f_i}}\ne -1$ with $a=c_{f_i}$. Consider the norm map

  $$N:{\mathbb{F}}_{q^{2f_i}}^{\times }\to {\mathbb{F}}_{q^{f_i}}^{\times },N\left(a\right)=a^{1+q^{f_i}}.$$

  The map is surjective, and each element of ${\mathbb{F}}_{q^{f_i}}^{\times }$ has exactly $q^{f_i}+1$ preimages. Thus, since $N\left(a\right)=-1$ admits $q^{f_i}+1$ solutions, the number of admissible a is

  $$(q^{2f_i}-1)-(q^{f_i}+1)=q^{2f_i}-q^{f_i}-2.$$

  By symmetry, the same enumeration applies when $c_{f_i}\in \langle u\rangle$ or $\langle v\rangle$.
• Case C: $c_{f_i}=a+bu$, with $a,b\in {\mathbb{F}}_{q^{2f_i}}$ and $b\ne 0$. Then,

  $$1+c_{f_i}^{1+q^{f_i}}=1+a^{1+q^{f_i}}+u{b}^{1+q^{f_i}}.$$

  The element is a unit exactly when $a^{1+q^{f_i}}\ne -1$, and b may be any non-zero element of ${\mathbb{F}}_{q^{2f_i}}$. Hence,

  $$\mathrm{Count}:(q^{2f_i}-q^{f_i}-2)(q^{2f_i}-1).$$

  By ring symmetry, the same count applies if $c_{f_i}=a+bv$.
• Case D: $c_{f_i}=a+bu+cv$. Again, $a^{1+q^{f_i}}\ne -1$, while $b,c\in {\mathbb{F}}_{q^{2f_i}}$ are arbitrary, giving

  $$\mathrm{Count}:(q^{2f_i}-q^{f_i}-2)q^{4f_i}.$$
• (ii) Enumeration for self-reciprocal components. Adding the disjoint counts from all cases yields

  $$1+3(q^{2f_i}-q^{f_i}-2)+2(q^{2f_i}-q^{f_i}-2)(q^{2f_i}-1)+(q^{2f_i}-q^{f_i}-2)q^{4f_i}.$$

  Simplifying gives

  $$q^{6f_i}-q^{5f_i}-2q^{3f_i}-6q^{2f_i}+5q^{f_i}+8.$$
• (iii) LCD condition for reciprocal pairs. Consider a pair

  $$({\mathcal{C}}_j^{\left(1\right)},{\mathcal{C}}_j^{\left(2\right)})=(\langle (1,c_{h_j}^{\left(1\right)})\rangle ,\langle (1,c_{h_j}^{\left(2\right)})\rangle ).$$

  The LCD criterion reads

  $$1+c_{h_j}^{\left(1\right)}{c}_{h_j}^{\left(2\right)}\in R_{q^{h_j},u,v}^{\times }.$$
• Case 1: $c_{h_j}^{\left(1\right)}=0$. Then, $1\in R^{\times }$; hence, $c_{h_j}^{\left(2\right)}$ may be arbitrary, giving $q^{3h_j}$ solutions.
• Case 2: $c_{h_j}^{\left(1\right)}\in R_{q^{h_j},u,v}^{\times }$. For each unit $x=c_{h_j}^{\left(1\right)}$, the condition is satisfied uniquely by

  $$c_{h_j}^{\left(2\right)}=-x^{-1}.$$

  Since $|R_{q^{h_j},u,v}^{\times }|=(q^{h_j}-1)q^{2h_j}$, the number of such pairs is

  $${(q^{h_j}-1)}^2{q}^{4h_j}.$$

  Combining both cases gives

  $$q^{3h_j}+{(q^{h_j}-1)}^2{q}^{4h_j}=q^{6h_j}-2q^{5h_j}+q^{4h_j}+q^{3h_j}.$$
• (iv) Final enumeration. Multiplying all contributions yields the total number of LCD double negacirculant codes,

  $${\mathcal{N}}_{\mathrm{dnega}}=\prod _{i=1}^e\left(q^{6f_i}-q^{5f_i}-2q^{3f_i}-6q^{2f_i}+5q^{f_i}+8\right)\prod _{j=1}^d\left(q^{6h_j}-2q^{5h_j}+q^{4h_j}+q^{3h_j}\right),$$

  as claimed. □

Example 2.

The factorization of $z^{10}+1$ over ${\mathbb{F}}_7$ decomposes into irreducible polynomials as

$$z^{10}+1=(z^2+1)(z^4+3z^3+4z^2+4z+1)(z^4+4z^3+4z^2+3z+1).$$

The quadratic factor $g_1\left(z\right)=z^2+1$ is self-reciprocal, satisfying $z^2{g}_1\left(\frac{1}{z}\right)=g_1\left(z\right)$ with parameter $f_1=1$ through degree correspondence $2=2f_1.$ The quartic factors form a reciprocal pair $(k_1,k_1^{\ast })$ where $k_1^{\ast }\left(z\right)=z^4{h}_1\left(\frac{1}{z}\right)$, yielding $h_1=4$. The degree summation formula is

$$n=\sum _{i=1}^{e}2f_i+2\sum _{j=1}^d{h}_j=2\left(1\right)+2\left(4\right)=10.$$

Applying Theorem 3 with parameters $q=7$, $s=1$, $t=1$ gives the unit count

$${\mathcal{N}}_{snega}=(7^1+1)7^2\times (7^4-1)7^8=392\times 2400\times 7^8,$$

where the first term corresponds to the self-reciprocal factor and the second to the reciprocal pair. Theorem 4 produces the constrained count

$${\mathcal{N}}_{dnega}=(7^6-7^5-2·7^3-6·7^2+43)\times (7^{24}-2·7^{20}+7^{16}+7^{12}),$$

with the sextic polynomial in $7^{f_i}$ capturing self-reciprocal constraints and the degree-24 expression reflecting reciprocal pair contributions.

Remark 2.

The example demonstrates how the factorization structure governs both counting formulas: self-reciprocal factors contribute through additive $(q^{f_i}+1)$ terms in Theorem 3 and higher-order vanishing conditions in Theorem 4, while reciprocal pairs appear multiplicatively via $(q^{h_j}-1)$ and through their characteristic polynomial relations. The degree verification ensures parameter consistency between the factorization and counting formulas.

4. Asymptotic Bounds for Self-Dual Double Circulant Codes

In this section, we establish explicit asymptotic existence bounds for self-dual and LCD double circulant codes over the local ring $R_{q,u,v}.$ The key ingredients are the exact enumeration results in Theorems 1 and 2, together with a sharp union-bound argument applied to the Gray images of these codes. Throughout this section, we assume that (i) n is an odd prime and (ii) q is a primitive root modulo n.

Lemma 3.

Let n be an odd prime and suppose that q is a primitive root modulo n. Then, the following statements hold in $R_{q,u,v}\left[z\right]$.

1. 

The polynomial $z^n-1$ factors are

$$z^n-1=(z-1)h\left(z\right),$$

where $h\left(z\right)$ is the unique Hensel lift of $h_0\left(z\right)=1+z+\cdots +z^{n-1}$ and is irreducible in $R_{q,u,v}\left[z\right]$.

2. 

The ideals $\langle z-1\rangle$ and $\langle h\left(z\right)\rangle$ are comaximal in $R_{q,u,v}\left[z\right]$.

3. 

Consequently, the following CRT decomposition holds:

$${\displaystyle \frac{R_{q,u,v}\left[z\right]}{\langle z^n-1\rangle }}\cong {\displaystyle \frac{R_{q,u,v}\left[z\right]}{\langle z-1\rangle }}\oplus {\displaystyle \frac{R_{q,u,v}\left[z\right]}{\langle h\left(z\right)\rangle }}.$$

(33)

Proof. 

Let $\pi :R_{q,u,v}\left[z\right]⟶{\mathbb{F}}_q\left[z\right]$ denote the natural reduction map modulo the maximal ideal $\mathfrak{m}=\langle u,v\rangle$. Since q is a primitive root modulo n and n is a prime, the cyclotomic factorization of $z^n-1$ over ${\mathbb{F}}_q$ is

$$z^n-1=(z-1)h_0\left(z\right),h_0\left(z\right)=1+z+\cdots +z^{n-1},$$

where $h_0\left(z\right)$ is irreducible and $gcd(z-1,h_0\left(z\right))=1$ in ${\mathbb{F}}_q\left[z\right]$.

• (1) Lifting the factorization and irreducibility. The ring $R_{q,u,v}$ is local with maximal ideal $\mathfrak{m}$ and residue field ${\mathbb{F}}_q$. By Hensel’s lemma for factorizations modulo nilpotent ideals: if $f\equiv g_0{h}_0\left(\mathrm{mod}\mathfrak{m}\right)$ with $gcd(g_0,h_0)=1$ in ${\mathbb{F}}_q\left[z\right]$, then there exist unique monic lifts $g_1,h_1\in R_{q,u,v}\left[z\right]$ satisfying $g_1\equiv g_0,h_1\equiv h_0\left(\mathrm{mod}\mathfrak{m}\right)$ and $f=g_1{h}_1$ in $R_{q,u,v}\left[z\right]$.

Applying this to $f\left(z\right)=z^n-1$, $g_0\left(z\right)=z-1$ and $h_0\left(z\right)$ as above, we obtain the lifted factorization

$$z^n-1=(z-1)h\left(z\right),\pi \left(h\right)=h_0.$$

To prove the irreducibility of $h\left(z\right)$, suppose that $h\left(z\right)=a\left(z\right)b\left(z\right)$ with non-unit $a,b\in R_{q,u,v}\left[z\right]$. Reducing modulo $\mathfrak{m}$ gives $h_0\left(z\right)=\pi \left(a\left(z\right)\right)\pi \left(b\left(z\right)\right)$ in ${\mathbb{F}}_q\left[z\right]$. Since $h_0\left(z\right)$ is irreducible, one of $\pi \left(a\right(z\left)\right)$, $\pi \left(b\right(z\left)\right)$ must be a non-zero constant in ${\mathbb{F}}_q$. Assume that $\pi \left(a\left(z\right)\right)=\omega \in {\mathbb{F}}_q^{\times }$. Then, $a\left(z\right)=\omega +\eta \left(z\right)$ with $\eta \left(z\right)\in \mathfrak{m}\left[z\right]$, and $\omega +\eta \left(z\right)$ is a unit in $R_{q,u,v}\left[z\right]$. Hence, $a\left(z\right)$ is a unit, which is a contradiction. Therefore, $h\left(z\right)$ is irreducible in $R_{q,u,v}\left[z\right]$.

• (2) Coprimeness. Since $gcd(z-1,h_0\left(z\right))=1$ in ${\mathbb{F}}_q\left[z\right]$, there exist $a_0\left(z\right),b_0\left(z\right)\in {\mathbb{F}}_q\left[z\right]$ such that

  $$a_0\left(z\right)(z-1)+b_0\left(z\right)h_0\left(z\right)=1.$$

  Interpreting $a_0$ and $b_0$ as elements of $R_{q,u,v}\left[z\right]$ gives

  $$a_0\left(z\right)(z-1)+b_0\left(z\right)h\left(z\right)=1+t\left(z\right),t\left(z\right)\in \mathfrak{m}\left[z\right].$$

  Since $1+t\left(z\right)$ is a unit in $R_{q,u,v}\left[z\right]$, multiplying both sides by its inverse yields

  $$a\left(z\right)(z-1)+b\left(z\right)h\left(z\right)=1$$

  for some $a\left(z\right),b\left(z\right)\in R_{q,u,v}\left[z\right]$. Thus, $\langle z-1\rangle +\langle h\left(z\right)\rangle =R_{q,u,v}\left[z\right],$ and the two ideals are coprime.
• (3) Chinese remainder decomposition. Since $(z-1)h\left(z\right)=z^n-1$ and the ideals $\langle z-1\rangle$ and $\langle h\left(z\right)\rangle$ are coprime,

  $$\langle z^n-1\rangle =\langle z-1\rangle \cap \langle h\left(z\right)\rangle .$$

  The CRT therefore gives the ring isomorphism

  $${\displaystyle \frac{R_{q,u,v}\left[z\right]}{\langle z^n-1\rangle }}\cong {\displaystyle \frac{R_{q,u,v}\left[z\right]}{\langle z-1\rangle }}\oplus {\displaystyle \frac{R_{q,u,v}\left[z\right]}{\langle h\left(z\right)\rangle }}.$$

□

A codeword c in a cyclic code $\mathcal{C}\subseteq R_{q,u,v}^n$ is said to be constant if it lies entirely in the component associated with the ideal $\langle z-1\rangle$ or, equivalently, if its projection belongs to the ideal $\langle h\left(z\right)\rangle \subset R_{q^{n-1},u,v}$. Constant codewords reflect the trivial part of the decomposition under the CRT and do not contribute to the exponential growth of the code family. In contrast, non-constant vectors arise from the $\langle h\left(z\right)\rangle$-component corresponding to the irreducible factor of degree $n-1$, and they are precisely the ones relevant for bounding the number of codes containing a given low-weight vector.

Remark 3.

The Gray map is an isometry that maintains orthogonality, so self-duality and LCD characteristics over $R_{q,u,v}$ exactly correspond to Euclidean self-duality (or LCD) over ${\mathbb{F}}_q$. While a double circulant code of length $2n$ possesses a Gray image of length $6n$, the CRT decomposition indicates that solely the non-constant component of length $3n$ influences the asymptotic expansion of the code family. Limiting the union-bound argument to this non-constant block produces precise and accurate asymptotic bounds, thereby encapsulating the influence of the Gray image length and the inner product structure.

Lemma 4.

Let $z=(y,\theta )\in R_{q,u,v}^{2n}$ be a non-zero vector such that y is non-constant. Then, the number of double circulant codes

$${\mathcal{C}}_x=\langle (1,x)\rangle \subset R_{q,u,v}^{2n}$$

that contain z satisfies the uniform bound

$$\#\left\{{\mathcal{C}}_x:z\in {\mathcal{C}}_x\right\}\le q^{2n+1}.$$

Proof. 

By the Chinese remainder decomposition given in Equation (33),

$${\displaystyle \frac{R_{q,u,v}\left[z\right]}{\langle z^n-1\rangle }}\cong R_{q,u,v}\oplus R_{q^{n-1},u,v},$$

every element (and, in particular, every generator x) splits uniquely as

$$x=(x_1,x_2),x_1\in R_{q,u,v},x_2\in R_{q^{n-1},u,v}.$$

Likewise, write $y=(y_1,y_2)$ and $\theta =({\theta }_1,{\theta }_2)$. The membership condition $z\in {\mathcal{C}}_x$ is equivalent to

$$\theta =y·x,$$

which decouples into the two componentwise equalities

$$\begin{array}{cc}\hfill {\theta }_1& =y_1{x}_1,\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill {\theta }_2& =y_2{x}_2.\hfill \end{array}$$

(35)

We count admissible $x=(x_1,x_2)$ by counting the solutions of (34) and (35) and multiplying.

• (i) Counting solutions to ${\theta }_1=y_1{x}_1$. Every element of $R_{q,u,v}$ has the unique form

  $$t=a+ub+vc,a,b,c\in {\mathbb{F}}_q,$$

  with multiplication rule

  $$(\alpha +u\beta +v\gamma )(a+ub+vc)=\alpha a+u(\alpha b+\beta a)+v(\alpha c+\gamma a)$$

  (since $u^2=v^2=uv=vu=0$). Write

  $$x_1=a_1+u{b}_1+v{c}_1,y_1={\alpha }_1+u{\beta }_1+v{\gamma }_1,{\theta }_1={\alpha }_1^{\prime }+u{\beta }_1^{\prime }+v{\gamma }_1^{\prime }.$$

  Then, (34) is equivalent to the scalar system over ${\mathbb{F}}_q$:

  $${\alpha }_1^{\prime }={\alpha }_1{a}_1,{\beta }_1^{\prime }={\alpha }_1{b}_1+{\beta }_1{a}_1,{\gamma }_1^{\prime }={\alpha }_1{c}_1+{\gamma }_1{a}_1.$$

We now enumerate the possibilities (for fixed $y_1,{\theta }_1$).

• If $y_1=0$, then ${\alpha }_1={\beta }_1={\gamma }_1=0$. In order for (34) to hold, we must have ${\theta }_1=0$ as well; in this case, no coordinate of $x_1$ is constrained, so there are $q^3$ choices for $x_1$.
• If $y_1$ is a non-zero scalar multiple of one of $1,u,v$ (i.e., exactly one of ${\alpha }_1,{\beta }_1,{\gamma }_1$ is non-zero), then exactly one of the three scalar equations above determines one coordinate of $x_1$, while the remaining two coordinates are free. Hence, there are $q^2$ solutions for $x_1$.
• If $y_1$ has two (or three) non-zero components (for example, $y_1={\alpha }_1+u{\beta }_1$ with ${\alpha }_1,{\beta }_1\ne 0$), then two of the scalar equations determine two coordinates of $x_1$ and the remaining coordinate is free, so there are q solutions for $x_1$.

• (ii) Counting solutions to ${\theta }_2=y_2{x}_2$. The same decomposition arguments apply in $R_{q^{n-1},u,v}$; write

  $$x_2=a_2+u{b}_2+v{c}_2,a_2,b_2,c_2\in {\mathbb{F}}_{q^{n-1}},$$

  and similarly for $y_2,{\theta }_2$. Equation (35) reduces to three scalar equalities over ${\mathbb{F}}_{q^{n-1}}$ of the same form as above. Because each free scalar now ranges over ${\mathbb{F}}_{q^{n-1}}$ (size $q^{n-1}$), the counts parallel the previous cases but with a larger field.

• $y_2=0$: This is impossible under the hypothesis that y is not constant, so this case is excluded.
• $y_2$ is a non-zero scalar multiple of $1,u$ or v: One coordinate of $x_2$ is determined and two are free, yielding $q^{2(n-1)}$ possibilities.
• $y_2$ has mixed form: Two coordinates of $x_2$ are determined and one is free, yielding $q^{n-1}$ possibilities.

• (iii) Combining the counts. For fixed z, the total number of admissible $x=(x_1,x_2)$ is the product of the numbers of choices for $x_1$ and for $x_2$. The maximum is attained when $x_1$ lies in the unconstrained subcase ($q^3$ choices) and $x_2$ lies in the subcase with two free coefficients ($q^{2(n-1)}$ choices); hence,

  $$\#\{{\mathcal{C}}_x\ni z\}\le q^3·q^{2(n-1)}=q^{2n+1}.$$

  This proves the lemma. □

Remark 4.

The bound is tight when $y_1=0$ and $y_2$ lies in the single-generator subspace, yielding the maximal solution space of size $q^{2n+1}$. This result plays a central role in the union-bound argument later, as it controls the number of self-dual or LCD codes containing any fixed low-weight non-constant vector.

Keeping the same notations, we have the following result.

Theorem 5.

Let $z=(y,\theta )\in R_{q,u,v}^{2n}$ be a non-zero vector such that y is non-constant. Then, the number of self-dual double circulant codes

$${\mathcal{C}}_x=\langle (1,x)\rangle \subset R_{q,u,v}^{2n}$$

that contain z satisfies the upper bound

$$\#\left\{\mathit{self}\text{-}\mathit{dual}{\mathcal{C}}_x:z\in {\mathcal{C}}_x\right\}\le 2\left(1+q^{\frac{n-1}{2}}\right)q^{n-1}.$$

Proof. 

The proof refines the counting argument of Lemma 4 by incorporating the self-duality constraint on the generator x.

• (i) Decomposition under CRT. By the Chinese remainder decomposition

  $${\displaystyle \frac{R_{q,u,v}\left[z\right]}{\langle z^n-1\rangle }}\cong R_{q,u,v}\oplus R_{q^{n-1},u,v},$$

  every generator x splits uniquely as $x=(x_1,x_2)$ with $x_1\in R_{q,u,v}$ (constant component) and $x_2\in R_{q^{n-1},u,v}$ (non-constant component). The membership condition $z\in {\mathcal{C}}_x$ is equivalent to

  $$\theta =y·x,$$

  which decouples into ${\theta }_1=y_1{x}_1$ and ${\theta }_2=y_2{x}_2$. The additional self-duality constraint acts independently on these two components.
• (ii) Constant component. For a self-dual double circulant code, the constant component $x_1$ must itself satisfy the self-duality condition at the local factor $R_{q,u,v}$. By Theorem 1, there are at most

  $$\#\left\{x_1\mathrm{admissible}\right\}\le 2$$

  such choices. The main counting effort therefore concerns $x_2$.
• (iii) Self-duality on the non-constant component. Self-duality of the entire code is equivalent to the self-duality of its non-constant component with respect to the involution induced by $z\mapsto z^{-1}$. Concretely, writing $x_2=a_2+u{b}_2+v{c}_2$ with $a_2,b_2,c_2\in {\mathbb{F}}_{q^{n-1}}$, the self-duality condition is

  $$1+x_2\overline{x_2}=1+x_2{x}_2^{q^{\frac{n-1}{2}}}=0,$$

  (36)

  where the bar denotes the automorphism induced by this involution; on coefficients, it acts as the Frobenius map $a\mapsto a^{q^{\frac{n-1}{2}}}.$ Expanding (36) gives the scalar system

  $$\begin{array}{cc}\hfill 1+a_2{a}_2^{q^{\frac{n-1}{2}}}& =0,\hfill \end{array}$$

  (37)

  $$\begin{array}{cc}\hfill a_2{b}_2^{q^{\frac{n-1}{2}}}+b_2{a}_2^{q^{\frac{n-1}{2}}}& =0,\hfill \end{array}$$

  (38)

  $$\begin{array}{cc}\hfill a_2{c}_2^{q^{\frac{n-1}{2}}}+c_2{a}_2^{q^{\frac{n-1}{2}}}& =0.\hfill \end{array}$$

  (39)
• (iv) Counting solutions to the self-duality equations. Proposition 1 gives the exact number of solutions to (37): there are at most $1+q^{\frac{n-1}{2}}$ admissible values of $a_2$. For each such $a_2$, Equations (38) and (39) impose linear relations on $b_2$ and $c_2$ over ${\mathbb{F}}_{q^{n-1}}$, leaving at most $q^{2(n-1)}$ degrees of freedom overall. Thus,

  $$\#\left\{x_2\mathrm{satisfying}\mathrm{self}\text{-}\mathrm{duality}\right\}\le (1+q^{\frac{n-1}{2}})q^{2(n-1)}.$$
• (v) Intersection with the membership constraint. The linear equation ${\theta }_2=y_2{x}_2$ further restricts $x_2$. Depending on the structure of $y_2$, some coordinates are fixed while others remain free (see the case analysis in Theorem 4). The maximal number of admissible $x_2$ occurs in the case with two free coordinates, giving

  $$\#\left\{x_2\mathrm{admissible}\right\}\le (1+q^{\frac{n-1}{2}})q^{n-1}.$$

  When fewer coordinates are free, this number is strictly smaller.
• (vi) Final multiplication. Combining the bound 2 for $x_1$ with the bound above for $x_2$ gives

  $$\#\left\{\mathrm{self}\text{-}\mathrm{dual}{\mathcal{C}}_x:z\in {\mathcal{C}}_x\right\}\le 2(1+q^{\frac{n-1}{2}})q^{n-1},$$

  which completes the proof. □

Remark 5.

The bound is attained in the extremal configuration where $x_1$ runs over all 2 admissible constant solutions, while $y_2$ yields two free coordinates in the non-constant component. This estimate is crucial for the union-bound argument in the next theorem, as it controls the number of self-dual codes containing any fixed low-weight non-constant vector.

Now, we are in a position to prove the following theorem.

Theorem 6.

Suppose that $\delta >0.$ Then, there exist infinite families of double circulant codes over R with length $2n$ and rate ${\displaystyle \frac{1}{2}}$ whose Gray images have a relative distance of at least δ. For self-dual codes, this holds when $\delta \in (0,\frac{1}{12})$, while, for LCD codes, it holds when $\delta \in (0,\frac{1}{6})$. Both families are asymptotically good.

Proof. 

Under the hypothesis of Lemma 3 and the notation of (20) in Theorem 1, there is the trivial linear factor $x-1$ (handled by the leading constant 2 in (20)). Moreover, there is one self-reciprocal factor h of degree $n-1=2f$, so $f=\frac{n-1}{2},$ and, since are no reciprocal pairs, $d=0$. Thus, $e=2$, and the only term in the product ${\prod }_{i=2}^e$ is the one coming from h with $f_2=\frac{n-1}{2}$; the product over j is empty. Plugging into (20), we obtain

$${\mathcal{N}}_s=2(q^{3f_2}+q^{2f_2})=2(q^{\frac{3(n-1)}{2}}+q^{n-1})=2q^{n-1}(q^{\frac{n-1}{2}}+1).$$

(40)

As $n\to \infty$, the dominant term is $2q^{\frac{3(n-1)}{2}}=2q^{\frac{3n-3}{2}}$. Hence,

$${\mathcal{N}}_s\sim 2q^{\frac{3n-3}{2}}.$$

For any $\delta ,$ we define the integer-valued weight threshold

$$\delta \left(n\right)=\lfloor \delta n\rfloor .$$

Note that $\delta \left(n\right)\le \delta n<\delta \left(n\right)+1$. The number of vectors $z\in R_{q,u,v}^{2n}$ with $\mathrm{wt}\left(z\right)\le \delta \left(n\right)$ is bounded by

$$\begin{array}{cc}\hfill B(n,\delta )& =\sum _{k=1}^{\delta \left(n\right)}\left({\displaystyle \genfrac{}{}{0pt}{}{3n}{k}}\right){(q-1)}^k\hfill \\ \hfill & \le (\delta \left(n\right)+1)\left({\displaystyle \genfrac{}{}{0pt}{}{3n}{\delta \left(n\right)}}\right){(q-1)}^{\delta \left(n\right)}.\hfill \end{array}$$

Using Stirling’s approximation,

$$\left({\displaystyle \genfrac{}{}{0pt}{}{3n}{\delta \left(n\right)}}\right)\le {\left({\displaystyle \frac{3ne}{\delta \left(n\right)}}\right)}^{\delta \left(n\right)}\le {\left({\displaystyle \frac{3e}{\delta }}\left(1+{\displaystyle \frac{1}{\delta n}}\right)\right)}^{\delta \left(n\right)},$$

where e is the Euler number. Thus,

$$B(n,\delta )\le (\delta n+1){\left({\displaystyle \frac{3e(q-1)}{\delta }}\left(1+{\displaystyle \frac{1}{\delta n}}\right)\right)}^{\delta \left(n\right)}.$$

By Theorem 5, each such low-weight vector belongs to at most

$$N^{\prime }=2(1+q^{\frac{n-1}{2}})q^{n-1}.$$

self-dual codes. To guarantee the existence of a code with no low-weight vectors, we require

$${\mathcal{N}}_{\mathrm{self}\text{-}\mathrm{dual}}>B\left(\delta n\right)·N^{\prime },$$

which becomes

$$2q^{\frac{3n-1}{2}}>\delta n{\left({\displaystyle \frac{3e(q-1)}{\delta }}\right)}^{\delta n}·(2(1+q^{\frac{n-1}{2}})q^{n-1}).$$

Simplifying and taking logarithms yields the key inequality

$${\displaystyle \frac{3}{2}}logq>{\displaystyle \frac{log\left(\delta n\right)}{n}}+{\displaystyle \frac{log(1+q^{\frac{n-1}{2}})}{n}}+\delta log\left({\displaystyle \frac{3e(q-1)}{\delta }}\right)+logq.$$

As $n\to \infty$, this reduces to

$${\displaystyle \frac{1}{2}}logq>\delta log\left({\displaystyle \frac{3e(q-1)}{\delta }}\right),$$

satisfied when $\delta <\frac{1}{12}$.

For LCD codes, plugging $f=\frac{n-1}{2}$ into Formula (24) yields

$$\begin{array}{cc}\hfill {\mathcal{N}}_d& =(q-2)q^2\left(\underset{q^{2f}-q^f-1}{\underbrace{q^{n-1}-q^{\frac{n-1}{2}}-1}}\right)\underset{q^{4f}}{\underbrace{q^{2(n-1)}}}\hfill \\ \hfill & =(q-2)q^{2n}\left(q^{n-1}-q^{\frac{n-1}{2}}-1\right)\hfill \\ \hfill & \left(q-2\right)q^{3n-1}.\hfill \end{array}$$

By ignoring the constant factor $(q-2)$ and the fixed $q^{-1}$ as $n\to \infty ,$ we have

$${\mathcal{N}}_{\mathrm{LCD}}\sim q^{3n}.$$

Now, each low-weight vector appears in at most $N^{\prime }=q^{2n+1}$ codes. The existence condition

$$q^{3n}>\delta n{\left({\displaystyle \frac{3e(q-1)}{\delta }}\right)}^{\delta n}{q}^{2n+1}$$

simplifies to

$$q^n>\delta nq{\left({\displaystyle \frac{3e(q-1)}{\delta }}\right)}^{\delta n}.$$

The asymptotic analysis leads to

$$1>\delta log\left({\displaystyle \frac{3e(q-1)}{\delta }}\right),$$

giving $\delta <\frac{1}{6}.$ □

Remark 6.

These calculations demonstrate that, for sufficiently large n, there exist codes in both families that meet the stated distance limits. The fixed rate $\frac{1}{2}$ and the positive relative distance establish the asymptotic goodness of these code families. Moreover, $\frac{1}{12}$ and $\frac{1}{6}$ emerge naturally from balancing the growth rates of the code counts and the low-weight vector bounds in the logarithmic inequalities.

Example 3.

For a double circulant code $C\subset R_{7,u,v}^{2n}$ with generator matrix $G=[I_n\mid M]$, where

$$M=M_1+u{M}_2+v{M}_3,$$

where each $M_i$ is an $n\times n$ circulant over ${\mathbb{F}}_7$, we evaluate the performance via the Gray image. Since ${\psi }_i:R_{7,u,v}^n\to {\mathbb{F}}_7^{3n}$ is a Lee–Hamming isometry, the minimum Lee distance of C equals the minimum Hamming distance of ${\psi }_i\left(C\right)$, whose block length is $3·\left(2n\right)=6n$ and whose dimension is $3n.$ Thus, ${\psi }_i\left(C\right)$ has a generator matrix of the form:

$$M^{\left(1\right)}={\left(\begin{array}{cccccc}0& 2I& 0& -M_2& 2M_1+M_2& M_3\\ -I& I& 0& -(M_1+M_2)& M_1+M_2& 0\\ 0& 0& I& 0& 0& M_1+M_3\end{array}\right)}_{3n\times 6n},$$

$$M^{\left(2\right)}={\left(\begin{array}{cccccc}I& I& I& M_1& M_1+M_2& M_1+M_3\\ 0& I& 0& 0& M_1+M_2& 0\\ 0& 0& I& 0& 0& M_1+M_3\end{array}\right)}_{3n\times 6n}.$$

In the tables (

Table 1. ${\mathbb{F}}_7$-Gray images of good families of double circulant codes of length n.

n	Gray Map	Coefficients	Type	Parameters
2	${\psi }_1$	$a_1=[3,1],a_2=[1,4],a_3=[5,3]$	LCD	${[12,6,3]}_7$
2	${\psi }_2$	$a_1=[2,0],a_2=[4,1],a_3=[0,2]$	Self-dual	${[12,6,3]}_7$
3	${\psi }_1$	$a_1=[1,4,0],a_2=[2,1,3],a_3=[5,0,1]$	Self-dual	${[18,9,4]}_7$
3	${\psi }_2$	$a_1=[3,2,1],a_2=[0,1,4],a_3=[6,3,0]$	Self-dual	${[18,9,1]}_7$
4	${\psi }_1$	$a_1=[1,0,2,3],a_2=[4,1,0,2],a_3=[0,3,1,5]$	Self-dual	${[24,12,4]}_7$
4	${\psi }_2$	$a_1=[2,1,3,0],a_2=[1,4,2,1],a_3=[3,0,1,2]$	Self-dual	${[24,12,4]}_7$
5	${\psi }_1$	$a_1=[1,2,0,4,3],a_2=[3,0,1,2,0],a_3=[5,1,3,0,2]$	LCD	${[30,15,5]}_7$
5	${\psi }_2$	$a_1=[2,3,1,0,4],a_2=[1,0,2,3,1],a_3=[0,4,1,2,3]$	LCD	${[30,15,10]}_7$
7	${\psi }_1$	$a_1=[1,3,0,2,4,1,5],a_2=[2,0,1,3,0,2,1],a_3=[3,1,2,0,1,3,0]$	LCD	${[42,21,5]}_7$
7	${\psi }_2$	$a_1=[4,2,1,3,0,2,1],a_2=[1,3,2,0,4,1,2],a_3=[2,1,0,4,3,0,1]$	LCD	${[42,21,13]}_7$
9	${\psi }_1$	$a_1=[1,3,0,2,4,1,5,0,2],a_2=[2,0,1,3,0,2,1,4,0],a_3=[3,1,2,0,1,3,0,2,1]$	LCD	${[54,27,5]}_7$

), we therefore report ${[6n,3n,d]}_{{\mathbb{F}}_7}$ parameters for ${\psi }_i\left(C\right)$; equivalently, C has length $2n$ over $R_{7,u,v}$ and Lee distance d. To specify the circulant M, we list the three seed polynomials $a_1\left(z\right),a_2\left(z\right),a_3\left(z\right)\in {\mathbb{F}}_7\left[z\right]$ that generate $M_1,M_2,M_3$ via the usual circulant construction. Vectors are written in coefficient order $[c_2,c_1,c_0]$, meaning $c_0+c_{1}z+c_2{z}^2$. For instance, the entry $a_1=[1,4,0]$ denotes $a_1\left(z\right)=z^2+4z$, so $M_1=\mathrm{circ}(c_0,c_1,\dots )$, with the first row built from the coefficients of $a_1\left(z\right)$ (and analogously for $a_2,a_3$). All distances d are obtained in Magma by constructing G, forming the ${\mathbb{F}}_7$-linear Gray image ${\psi }_i\left(C\right)\subset {\mathbb{F}}_7^{6n}$, and computing its minimum Hamming distance; by isometry, this equals the Lee distance of C. This convention is used for every entry in Table 1.

4.1. Discussion and Connection with Asymptotic Bounds

The numerical results presented in Table 1 illustrate finite-length instances of the asymptotic behavior established in Theorem 6. Each double circulant code over $R_{7,u,v}$ of length $2n$ yields, through the Gray isometry ${\psi }_i$, a 7-ary linear code of parameters ${[6n,3n,d]}_7$ and rate $R=\frac{1}{2}$. As n increases, the relative distance $\delta =\frac{d}{6n}$ of the Gray images remains strictly positive, confirming the asymptotic goodness predicted by Theorem 6.

In particular, the table shows that both self-dual and LCD constructions achieve a balanced rate and distance even for short lengths; for example, the self-dual code at $n=4$ attains parameters ${[24,12,4]}_7$ with $\delta =\frac{1}{6}$, while the LCD code at $n=5$ reaches ${[30,15,10]}_7$ with $\delta =\frac{1}{3}$. These finite examples lie within the theoretical distance intervals

$$\delta \in (0,\frac{1}{12})\mathrm{for}\mathrm{self}\text{-}\mathrm{dual}\mathrm{families},\delta \in (0,\frac{1}{6})\mathrm{for}\mathrm{LCD}\mathrm{families},$$

thereby concretely demonstrating the existence of infinite sequences of double circulant codes over $R_{q,u,v}$ whose Gray images over ${\mathbb{F}}_q$ are asymptotically good in the sense of having a non-zero rate and positive relative distance. Hence, the explicit constructions in Table 1 not only validate the theoretical estimates but also exemplify the practical realizability of the asymptotic distance thresholds.

4.2. Optimality Analysis

A linear ${[n,k,d]}_q$ code is called optimal if no other linear code with the same length n and dimension k achieves a strictly larger minimum distance $d^{\prime }>d$. Based on the parameters in Table 1, several of the constructed Gray images are either optimal or near-optimal with respect to the best-known ${\mathbb{F}}_7$ linear codes. In particular, the self-dual double circulant code of parameters ${[24,12,4]}_7$ and the LCD code ${[30,15,10]}_7$ both achieve the best-known distances for their lengths and dimensions; hence, they are optimal. Similarly, the LCD code ${[42,21,13]}_7$ is within one unit of the optimal distance bound ($d_{\mathrm{opt}}=14$), qualifying as near-optimal. These results confirm that the proposed double circulant constructions over $R_{7,u,v}$ not only align with the asymptotic predictions of Theorem 6 but also yield practically excellent codes that meet or approach the theoretical distance limits at finite lengths.

5. Conclusions

This study expands the framework of double circulant and double negacirculant codes to the local ring $R_{q,u,v}={\mathbb{F}}_q+u{\mathbb{F}}_q+v{\mathbb{F}}_q,u^2=v^2=uv=vu=0,$ which presents greater algebraic complexity than previously examined semi-local or chain rings. Explicit necessary and sufficient conditions for these codes to be self-dual or LCD were derived, and precise enumeration formulas were established by solving key norm and trace equations over ${\mathbb{F}}_q$ 

Table 2. Summary of main results for double circulant and double negacirculant codes over $R_{q,u,v}$.

Component	Type	Degree	Ring	Self-Dual	LCD
Double Circulant: $z^n-1=\delta (z-1){\prod }_{i=1}^e{l}_i\left(z\right){\prod }_{j=1}^d{k}_j\left(z\right)k_j^{\ast }\left(z\right)$
$(z-1)$	Constant	1	$R_{q,u,v}$	2	$(q-2)q^2$
$l_i\left(z\right)$	Self-recip.	$2f_i$	$R_{q^{2f_i},u,v}$	$q^{2f_i}+q^{3f_i}$	$(q^{2f_i}-q^{f_i}-1)q^{4f_i}$
$k_j\left(z\right)k_j^{\ast }\left(z\right)$	Conj. pair	$h_j$	$R_{q^{h_j},u,v}^2$	$(q^{h_j}-1)q^{2h_j}$	$q^{3h_j}+{\left[(q^{h_j}-1)q^{h_j}\right]}^2$
Total				$2{\prod }_{i=1}^e(q^{2f_i}+q^{3f_i})$	$(q-2)q^2{\prod }_{i=1}^e(q^{2f_i}-q^{f_i}-1)q^{4f_i}$
				$\times {\prod }_{j=1}^d(q^{h_j}-1)q^{2h_j}$	$\times {\prod }_{j=1}^d\left[q^{3h_j}+{\left[(q^{h_j}-1)q^{h_j}\right]}^2\right]$
Asymptotic (as $n\to \infty$)				$\sim 2q^{\frac{3n-1}{2}}$	$\sim q^{3n}$
Double Negacirculant: $z^n+1=\gamma {\prod }_{i=1}^e{l}_i\left(z\right){\prod }_{j=1}^d{k}_j\left(z\right)k_j^{\ast }\left(z\right)$
$l_i\left(z\right)$	Self-recip.	$2f_i$	$R_{q^{2f_i},u,v}$	$(q^{f_i}+1)q^{2f_i}$	$q^{6f_i}-q^{5f_i}-2q^{3f_i}-6q^{2f_i}+5q^{f_i}+8$
$k_j\left(z\right)k_j^{\ast }\left(z\right)$	Conj. pair	$h_j$	$R_{q^{h_j},u,v}^2$	$(q^{h_j}-1)q^{2h_j}$	$q^{6h_j}-2q^{5h_j}+q^{4h_j}+q^{3h_j}$
Total				${\prod }_{i=1}^e(q^{f_i}+1)q^{2f_i}$	${\prod }_{i=1}^e\left(q^{6f_i}-q^{5f_i}-2q^{3f_i}-6q^{2f_i}+5q^{f_i}+8\right)$
				$\times {\prod }_{j=1}^d(q^{h_j}-1)q^{2h_j}$	$\times {\prod }_{j=1}^d{q}^{6h_j}-2q^{5h_j}+q^{4h_j}+q^{3h_j}]$

. Our analysis establishes explicit bounds on the minimum distance of the Gray images, derived from the non-constant Gray block of length $3n$. This yields sharp and rigorously justified asymptotic distance estimates, ensuring that the resulting ${\mathbb{F}}_q$-linear images are distance-preserving and asymptotically good. Empirical evidence over ${\mathbb{F}}_7$ confirmed the theoretical predictions and illustrated that the suggested constructions produce effective linear codes with a balanced rate and minimum distance.

This study expands the comprehension of structured code families within local rings and emphasizes the significance of non-chain, nilpotent extensions in coding theory. Future research should focus on generalizing the existing enumeration and construction techniques to encompass multi-generator double circulant and quasi-cyclic codes over $R_{q,u,v}$ and associated local rings with higher-order nilpotent structures. Identifying equivalent algebraic conditions for self-duality, LCD behavior, and asymptotic optimality within these broader categories is an unresolved and promising issue.