Homogeneous Weights of Linear Codes over 𝔽_p^m[u, v] of Order p^4m

Abstract

Let $R={\mathbb{F}}_{p^m}[u,v]$ be the ring whose order is $p^{4m}$, where p denotes a prime number. The algebraic structure of R is substantially influenced by the relationships among the elements $u^2$, $uv$, and $v^2$. We establish that R is formed over ${\mathbb{F}}_{p^m}$ when $p\ne 2$ and $u^2=\gamma v^2$, where $\gamma$ is a non-zero element of ${\mathbb{F}}_{p^m}$ and $uv=0.$ For $p=2,$ the situations are those where $u^2=v^2=0$ and $uv\ne 0.$ After that, we discuss how to make matrices that are linked to the symmetrical weight enumerators of linear codes over these rings. These matrices are essential in the examination of coding theory over Frobenius rings. Utilizing these matrices makes it possible to determine the homogeneous distances of linear codes over the ring R. Our results show a connection between the uniform distances of linear codes and the matrices of symmetrized weight enumerators. This helps us understand the behavior and characteristics of these codes.

1. Introduction

In the initial stages of coding theory, codes were primarily examined over finite fields, particularly emphasizing the field of order two, leading to the creation of binary codes. In the past decade, there has been a notable surge in interest in linear codes over rings, as well as in the investigation of Gray maps that convert these codes into codes over finite fields. A significant work conducted by Sloane, Calderbank, and their colleagues in 1994 demonstrated that the Delsarte–Goethals codes, Kerdock codes, and Preparata codes can be derived as the Gray images of linear codes over ${\mathbb{Z}}_4.$ When a finite commutative ring R is considered as an R-module, R is called Frobenius if and only if $\mathrm{soc}\left(R\right)\cong {\mathbb{F}}_{p^m}$, where ${\mathbb{F}}_{p^m}={\displaystyle \frac{R}{J}},$ J is the Jacobson radical of R, and soc$\left(R\right)$ is defined later. Two old theorems by MacWilliams, the extension theorem and the MacWilliams identities, have recently been used to show that local Frobenius rings can be used as alphabets for linear codes. Just as these theorems are applicable to finite fields, they also hold for local Frobenius rings. This advancement not only broadened the domain of coding theory but also offered novel insights into the intricate structure and prospective applications of codes over rings; we refer to [1,2,3,4,5].

Suppose p is prime and m is a positive integer. A linear code C of length n over a ring R is defined as an R-submodule of $R^n$. The rings primarily analyzed in this context are those of the kind ${\mathbb{F}}_{p^m}[u,v]$, which have an order of $p^{4m}.$ The ring ${\mathbb{F}}_{p^m}[u,v]$ consists of all polynomials in the indeterminates u and v over the field ${\mathbb{F}}_{p^m}$. The radical J of R is the ideal formed by u, v, and w, with an order of $p^{3m}$. The residue field of R is defined as

$${\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{J}}\cong {\mathbb{F}}_{p^m}$$

of order $p^m$. Consequently,

$$R={\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}+v{\mathbb{F}}_{p^m}+w{\mathbb{F}}_{p^m},$$

with the ideal J represented as follows:

$$J=u{\mathbb{F}}_{p^m}+v{\mathbb{F}}_{p^m}+w{\mathbb{F}}_{p^m}.$$

The algebraic structure of R can be written as ${\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{I}}$, where I is an ideal made up of $u^2$, $uv$, and $v^2$, as we will see soon. Assume that C is a linear code over R with length n. A weight $\omega$ is defined as a function that assigns non-negative integers to elements of the ring. If $\mathbf{c}$ is a codeword, then the weight of $\mathbf{c}$ is defined in terms of the weights of all its coordinates, thus extending this function to the code. For a codeword $\mathbf{c}=(c_1,c_2,\dots ,c_n)\in C$, the weight is

$$w\left(\mathbf{c}\right)=\omega \left(c_1\right)+\omega \left(c_2\right)+\cdots +\omega \left(c_n\right).$$

In coding theory, the predominant weight utilized is the Hamming weight, which assigns a weight of zero to the zero coordinate and a weight of one to all non-zero coordinates. We represent this weight as ${\omega }_H$, ensuring that ${\omega }_H\left(0\right)=0$ and ${\omega }_H\left(x\right)=1$ for all $x\ne 0$. Additionally, the authors of [2,3,4] present the Lee weight defined on ${\mathbb{Z}}_4$ as an important weight. This weight is denoted by $w_L,$ and its formula is as follows:

$${\omega }_L\left(x\right)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}x=0,\hfill \\ 2,\hfill & \mathrm{if}x=2,\hfill \\ 1,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(1)

A generalization of ${\omega }_L$ to R is referred to as the homogeneous weight, which we denote by ${\omega }_{hom}$. The $\omega$-weight enumerator of C is the polynomial

$$P_C\left(z\right)=\sum _{\mathbf{c}\in C}{z}^{\omega \left(\mathbf{c}\right)}.$$

(2)

In coding theory, the predominant weight utilized is the Hamming weight, which assigns a weight of zero to the zero coordinate and a weight of one to all non-zero coordinates. We represent this weight as ${\omega }_H$, ensuring that ${\omega }_H\left(0\right)=0$ and ${\omega }_H\left(x\right)=1$ for all $x\ne 0$. Additionally, the authors of [2,3,4] present the Lee weight defined on ${\mathbb{Z}}_4$ as an important weight. This weight is denoted by $w_L,$ and its formula is as follows. Let $R={\mathbb{F}}_{p^m}[u,v]$ with an order of $p^{4m}$. This study seeks to accomplish two objectives. Initially, we seek to formulate the matrices S that correspond to the symmetrized weight enumerators. Secondly, we seek to ascertain the homogeneous weights of these codes using the second columns of the matrices S.

The study’s structure is as follows: Section 2 presents a review of essential topics pertaining to ${\mathbb{F}}_{p^m}[u,v]$ and their characteristics. In Section 3, we conclude our examination of all local Frobenius rings $R={\mathbb{F}}_{p^m}[u,v]$ that comprise $p^{4m}$ elements. Section 4 delineates the matrices S associated with the symmetrized weight enumerators. Finally, Section 5 presents the main results of this research by discussing the weights that are all the same in the given ring R. The article also presents other examples that illustrate the principal findings.

2. Preliminaries

This part presents the fundamental notation and fact utilized in the investigation. In the following, let J be the Jacobson radical of R, which is a finite local ring. Furthermore, $R\left[x\right]$ contains all polynomials over R, while $\langle y\rangle$ symbolizes the ideal formed by y. For the results reported above, see [6,7,8,9,10,11,12].

It is important to note that a ring R characterized by a unique maximal ideal J is classified as a local ring. Consequently, it can be deduced that ${\displaystyle \frac{R}{J}}={\mathbb{F}}_{p^m}\cong \mathrm{GF}\left(p^m\right)$. A finite local ring R can be classified as Frobenius when its only minimal ideal is ${\mathrm{ann}}_R\left(J\right)$, the annihilator of J, which is cyclic. In equivalent terms, R is Frobenius if ${\displaystyle \frac{R}{J}}\cong \mathrm{soc}\left(R\right)$, with $\mathrm{soc}\left(R\right)$ representing the direct sum of all minimal ideals of R. In local Frobenius rings, it has been demonstrated that ${\mathrm{ann}}_R\left(J\right)=\mathrm{soc}\left(R\right)=J^{t-1}$, with t representing the index of nilpotency of J [7,13]. Our discussion focuses on $R={\mathbb{F}}_{p^m}[u,v]$ of order $p^{4m}$. This suggests that

$$R={\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}+v{\mathbb{F}}_{p^m}+w{\mathbb{F}}_{p^m},$$

(3)

where w is either $u^2$ or $uv$, as we will see in the next section. R is a local ring with $J=\langle u,v\rangle$ and is not a chain ring. To maintain clarity, we will define $\xi$ as a primitive element of ${\mathbb{F}}_{p^m}$ and

$$\begin{array}{cc}& \Delta \left(m\right)=\left(\xi \right)\cup \left\{0\right\}=\{0,1,\xi ,{\xi }^2,\cdots ,{\xi }^{p^m-2}\};\hfill \\ & A=\{{\xi }^{2i+1}:1\le i\le {\displaystyle \frac{p^m-1}{2}}-1\}=\{\xi \in {\Delta }^{*}\left(m\right):\xi \notin {\left({\Delta }^{*}\left(m\right)\right)}^2\};\hfill \\ & B=\{{\xi }^{2i}:1\le i\le {\displaystyle \frac{p^m-1}{2}}-1\}=\{\xi \in {\Delta }^{*}\left(m\right):\xi \in {\left({\Delta }^{*}\left(m\right)\right)}^2\}.\hfill \end{array}$$

Since R is the sum of ${\mathbb{F}}_{p^m}$ by the elements $u,v,$ and $w,$ this means that every element x of R can be uniquely written as

$$r={\alpha }_0+u{\alpha }_1+v{\alpha }_2+w{\alpha }_3,$$

(4)

where $={\alpha }_i\in \Delta \left(m\right)$ for $i=0,1,2,3.$ Hence, the Frobenius map $\sigma$ and trace map Tr for R are defined by

$$\begin{array}{cc}\hfill & \sigma \left(r\right)={\alpha }_0^p+u{\alpha }_1^p+v{\alpha }_2^p+w{\alpha }_3^p,\hfill \\ \hfill & Tr\left(r\right)=r+\sigma \left(r\right)+{\sigma }^2\left(r\right)+\cdots +{\sigma }^{m-1}\left(r\right).\hfill \end{array}$$

(5)

A linear code C of length n over R can be defined as an R-submodule of $R^n$. Consider a linear code C characterized by its length n and defined over the set R. To maintain clarity, we will define $\xi$ as a primitive element of ${\mathbb{F}}_{p^m}$, and it can be transformed through permutation equivalence into a code represented by the following generator matrix:

where $T_{ij}$ are matrices of various sizes.

The set of complex-valued functions defined on any finite Abelian group G is universally acknowledged to have a basis constituted by characters, which are homomorphisms. $G\to {\mathbb{C}}^{*}$. In the context of the additive group of $R,$ it is evident that any weight, even homogeneous weight, can be represented as a linear combination of characters. The characteristics of $(R,+)$ are expressed as $\vartheta :x\mapsto {\zeta }^{\mathrm{Tr}\left(ax\right)}$, where a ranges across R. Let ∼ be a relation defined on R such that $x\sim y$ if there exists $a\in U\left(R\right)$ for which $x=ay$. This signifies that the unit group of R operates by translation on the set R, thereby establishing an equivalence relation. We denote the equivalence classes as ${\widehat{d}}_1,{\widehat{d}}_2,\cdots ,{\widehat{d}}_q$ and define $n_i^{\prime }\left(\mathbf{c}\right)$ as the count of items in ${\widehat{b}}_i$ present in the codeword $\mathbf{c}$. The symmetrized weight enumerator (SWE) is given by

$$SW{E}_C(x_{{\widehat{s}}_1},\cdots ,x_{{\widehat{s}}_q})=\sum _{\mathbf{c}\in C}\prod _i{x_{{\widehat{s}}_i}}^{n_i^{\prime }\left(\mathbf{c}\right)}(\mathrm{symmetrized}\mathrm{weight}\mathrm{enumerator}).$$

(6)

The MacWilliams equation for the SWE is as follows:

$$SW{E}_C(x_{{\widehat{d}}_1},\cdots ,x_{{\widehat{d}}_q})={\displaystyle \frac{1}{\mid C^{\perp }\mid }}SW{E}_{C^{\perp }}(S·(x_{{\widehat{d}}_1},\cdots ,x_{{\widehat{d}}_q})),$$

(7)

where $S=\left(d_{ij}\right)$ and

$$d_{ij}=\sum _{a\in {\widehat{d}}_j}\vartheta \left(d_{i}a\right).$$

3. On the Ring ${\mathbb{F}}_{{\mathit{p}}^{\mathit{m}}}[\mathit{u},\mathit{v}]$ with Order ${\mathit{p}}^{\mathbf{4}\mathit{m}}$

Let R denote a local Frobenius ring defined as $R={\mathbb{F}}_{p^m}[u,v]$ with an order of $p^{4m}$ from this point onward. By altering the relationships among $u^2,uv$, and $v^2$, this section will elucidate the construction of R and their numbers. Considering that $\left|R\right|=p^{4m}$, R can be represented as an ${\mathbb{F}}_{p^m}$-module of

$$R={\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}+v{\mathbb{F}}_{p^m}+w{\mathbb{F}}_{p^m},$$

(8)

where, as we will show later,

$$w=\left\{\begin{array}{cc}{u}^2,\hfill & \mathrm{if}p\ne 2,\hfill \\ uv,\hfill & \mathrm{if}p=2.\hfill \end{array}\right.$$

The unit group of R, denoted as $U\left(R\right)$, plays a significant role in the process of deriving the construction of R. The structure of $U\left(R\right)$ is given in [14] as follows:

$$U\left(R\right)=\langle \xi \rangle \times H,H=1+J.$$

Every unit x of R can be expressed as

$$x={\xi }_0+{\xi }_{1}u+{\xi }_{2}v+{\xi }_{3}w,$$

where ${\xi }_0\in {\Delta }^{*}\left(m\right)$ and ${\xi }_i\in \Delta \left(m\right),$$i=1,2,3.$

Theorem 1.

Let $R={\mathbb{F}}_{p^m}[u,v]$ with order $p^{4m}.$ Then,

$$R\cong \left\{\begin{array}{c}\left\{\begin{array}{c}{\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2-v^2,uv\rangle }},\hfill \\ {\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2-\xi v^2,uv\rangle }},\hfill \end{array}\right.(\mathit{if}p\ne 2)\hfill \\ \\ \left\{\begin{array}{c}{\displaystyle \frac{{\mathbb{F}}_{2^m}[u,v]}{\langle u^2-v^2,uv\rangle }},\hfill \\ {\displaystyle \frac{{\mathbb{F}}_{2^m}[u,v]}{\langle u^2,v^2\rangle }}.\hfill \end{array}\right.(\mathit{if}p=2)\hfill \end{array}\right.$$

Proof. 

Since the prime ring of R is ${\mathbb{F}}_{p^m}$ and the order is $p^{4m},$ the number of generators of J is three, and it follows that the dimension of ${\displaystyle \frac{J}{J^2}}$ over ${\mathbb{F}}_{p^m}$ is two and that of $J^2$ is one. Consequently, the generators of J are u, v, and w, leading to the following structure:

$$R={\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}+v{\mathbb{F}}_{p^m}+w{\mathbb{F}}_{p^m},$$

with the ideal J represented as

$$J=u{\mathbb{F}}_{p^m}+v{\mathbb{F}}_{p^m}+J^2,$$

and

$$J^2=w{\mathbb{F}}_{p^m},$$

where w can be chosen as $u^2$, $v^2$, or $uv$. Note that the cases where $w=u^2$ and $w=v^2$ are equivalent, yielding only two distinct scenarios. If $uv=0$, then we set $w=u^2$, leading to the equation

$$u^2=\gamma v^2,$$

where $\gamma \in {\Delta }^{*}\left(m\right)$. Specifically, if $\gamma =0$, then $u^2=0$, and so $\mathrm{soc}\left(R\right)=\langle u,v\rangle$, indicating that R is not Frobenius. If $p\ne 2$ and $\gamma \in B$, then, applying the mapping $u\mapsto u$ and $v\mapsto \sqrt{\gamma }v$, we conclude that $u^2=v^2$. Conversely, if $\gamma \in A$, i.e., $\gamma =\xi$, we obtain $u^2=\xi v^2$. For the case when $p=2$, we generally have $\gamma \in B$, leading to $u^2=v^2$. Now, consider when $uv\ne 0$. If either $u^2=\gamma uv$ or $v^2=\gamma uv$ with $\gamma \ne 0$, we can transform the variables by letting $u\mapsto u$ and $v\mapsto u-\gamma v$, or alternatively, $u\mapsto v-\gamma u$ and $v\mapsto v$, thus reverting to the initial case. Supposing $u^2=v^2=0$, we perform the transformations $u\mapsto u+v$ and $v\mapsto u-v$ when $p\ne 2$, which simplifies the problem to the situation where $uv=0$. However, for $p=2$, this simplification does not hold because $2\notin U\left(R\right)$. Here, all elements of $\langle u,v\rangle$ in ${\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2,v^2\rangle }}$ have square zero, while the elements u and v in ${\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2-v^2,uv\rangle }}$ satisfy $u^2=v^2\ne 0$. Therefore, we conclude that

$${\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2,v^2\rangle }}\neg \cong {\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2-v^2,uv\rangle }}.$$

Thus,

$$\left\{\begin{array}{cc}\left\{\begin{array}{c}{u}^2=v^2,\hfill \\ u^2=\xi v^2,\hfill \end{array}\right.\hfill & uv=0\mathrm{and}(p\ne 2)\hfill \\ \\ \left\{\begin{array}{c}{u}^2=v^2\mathrm{and}uv=0,\hfill \\ u^2=v^2=0\mathrm{and}uv\ne 0.\hfill \end{array}\right.\hfill & (p=2)\hfill \end{array}\right.$$

This leads us to identify two rings characterized by

$$\left\{\begin{array}{c}{\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2-v^2,uv\rangle }},\hfill \\ {\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2-\xi v^2,uv\rangle }}.\hfill \end{array}\right.$$

These rings cannot be isomorphic when $p\ne 2$ because, in that case, assuming isomorphism would imply $\xi \in B$, contradicting the fact that $\langle \xi \rangle ={\mathbb{F}}_{p^m}^{*}$. However, an isomorphism

$${\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2-v^2,uv\rangle }}\cong {\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2-\xi v^2,uv\rangle }}$$

would require $p=2$. Assuming they are isomorphic, it follows that there are elements ${\delta }_1,{\delta }_2,{\delta }_3,{\delta }_4\in \Delta \left(m\right)$ such that $u_1={\delta }_{1}u+{\delta }_{2}v$ and $v_1={\delta }_{3}u+{\delta }_{4}v$ serve to span J. These generators preserve the relations of u and v, leading to equations of the form ${\gamma }_1^2+{\gamma }_2^2=\xi ({\gamma }_3^2+{\gamma }_4^2)$ with ${\gamma }_3^2+{\gamma }_4^2\ne 0$ and ${\gamma }_1{\gamma }_3+{\gamma }_2{\gamma }_4=0$, which only hold true if $p=2$. Let us take $p\ne 2$ and $uv\ne 0$. By making the substitutions $u⟼u+v$ and $v⟼u-v$ and following through with necessary calculations, we arrive at the conclusion that $u^2=\gamma v^2$, where $\gamma =-1$. This brings us back to the initial case, hence leading to either

$${\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2-v^2,uv\rangle }}\cong {\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2,v^2\rangle }}$$

or

$${\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2-\xi v^2,uv\rangle }}\cong {\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2,v^2\rangle }}$$

depending on whether $\gamma$ lies in B or $A.$ Assuming $p=2,$ we observe

$$\left\{\begin{array}{c}{\displaystyle \frac{{\mathbb{F}}_{2^m}[u,v]}{\langle u^2-v^2,uv\rangle }},\hfill \\ {\displaystyle \frac{{\mathbb{F}}_{2^m}[u,v]}{\langle u^2,v^2\rangle }}.\hfill \end{array}\right.$$

As established earlier, these two rings are in different classes. Consequently, we conclude that there are four distinct rings (where $t=3$ and $n=1$):

$$\left\{\begin{array}{cc}\left\{\begin{array}{c}{\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2-v^2,uv\rangle }},\hfill \\ {\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2-\xi v^2,uv\rangle }},\hfill \end{array}\right.\hfill & (\mathrm{if}p\ne 2)\hfill \\ \\ \left\{\begin{array}{c}{\displaystyle \frac{{\mathbb{F}}_{2^m}[u,v]}{\langle u^2-v^2,uv\rangle }},\hfill \\ {\displaystyle \frac{{\mathbb{F}}_{2^m}[u,v]}{\langle u^2,v^2\rangle }}.\hfill \end{array}\right.\hfill & (\mathrm{if}p=2)\hfill \end{array}\right.$$

□

Example 1.

Let $R={\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^3,v^2,uv\rangle }}$. Since $v\in \mathrm{ann}\left(J\right)$, it follows that R is not Frobenius because we have $\mathrm{ann}\left(J\right)=\langle u^2,v\rangle .$

Corollary 1.

There are two (non-isomorphic) rings when $p\ne 2$ and two rings if $p=2.$

Corollary 2.

The socle of R in Theorem 1 is $\langle u^2\rangle$ except for

$$\begin{array}{c}\hfill \mathrm{soc}\left({\displaystyle \frac{{\mathbb{F}}_{2^m}[u,v]}{\langle u^2,v^2\rangle }}\right)=\langle uv\rangle .\end{array}$$

This example illustrates the application of these rings in the construction of linear codes. In general, linear codes of length n over a ring R are characterized as R-submodules of $R^n$. In particular, cyclic codes of length n over R correspond to ideals of the quotient ring ${\displaystyle \frac{R\left[x\right]}{\langle x^n-1\rangle }}$. For more in-depth historical insights into linear codes over finite rings, refer to [6,15,16,17].

Example 2.

Consider the ring $R={\displaystyle \frac{{\mathbb{F}}_{2^4}[u,v]}{\langle u^2,v^2\rangle }}$. By Theorem 4, R is a local Frobenius ring with order $2^{16}$, and the set $\{u,v\}$ forms a minimal generating set for J, the maximal ideal of R, where $J^2=\langle uv\rangle$ and $J^3=\langle 0\rangle$. Additionally, $\Delta \left(4\right)={\mathbb{F}}_{16}$. Because R is a local Frobenius ring, its ideals are arranged in the following chain:

$$\langle 0\rangle =J^3\subset J^2=\langle uv\rangle \subset J=\langle u,v\rangle \subset R.$$

The proper ideals of R are structured in chains of the following form:

$$\begin{array}{cc}\hfill & \langle 0\rangle \subset \langle uv\rangle \subset \langle u\rangle \subset J,\hfill \\ \hfill & \langle 0\rangle \subset \langle uv\rangle \subset \langle v\rangle \subset J,\hfill \\ \hfill & \langle 0\rangle \subset \langle uv\rangle \subset \langle \alpha u+v\rangle \subset J,(\ast )\hfill \\ \hfill & \langle 0\rangle \subset \langle uv\rangle \subset \langle \alpha v+u\rangle \subset J,\hfill \end{array}$$

where $\alpha \in \Delta \left(4\right).$ If we take $s=7,$ then $x^7-1=(x+1)(x^3+x+1)(x^3+x^2+1)$ over ${\mathbb{F}}_{16},$ and hence, by Hensel’s lemma, the polynomial $x^7-1$ factors over R as

$$x^7-1=q_1\left(x\right)q_2\left(x\right)q_3\left(x\right).$$

Thus, we obtain (as a direct sum)

$${\displaystyle \frac{R\left[x\right]}{\langle x^7-1\rangle }}\cong {\displaystyle \frac{R\left[x\right]}{\langle q_1\left(x\right)\rangle }}+{\displaystyle \frac{R\left[x\right]}{\langle q_2\left(x\right)\rangle }}+{\displaystyle \frac{R\left[x\right]}{\langle q_3\left(x\right)\rangle }}$$

by the Chinese remainder theorem. Let $R_1=R={\displaystyle \frac{R\left[x\right]}{\langle q_1\left(x\right)\rangle }}$, $R_2={\displaystyle \frac{R\left[x\right]}{\langle q_2\left(x\right)\rangle }}$, and $R_3={\displaystyle \frac{R\left[x\right]}{\langle q_3\left(x\right)\rangle }}$. Additionally, define

$$\mathsf{\Gamma }\left(3\right)=\{{\alpha }_0+{\alpha }_{1}x+{\alpha }_2{x}^2:{\alpha }_i\in \Delta \left(4\right)\}$$

for the rings $R_2$ and $R_3$. The ideal lattices of $R_2$ and $R_3$ resemble that of R in (*), with $\alpha \in \mathsf{\Gamma }\left(3\right)$.

If C is a cyclic code over R with length $=7$, then C is an ideal of ${\displaystyle \frac{R\left[x\right]}{\langle x^7-1\rangle }}$ and can be expressed as a direct sum:

$$C\cong C_1+C_2+C_3,$$

where $C_1$, $C_2$, and $C_3$ are cyclic codes over R, $R_1$, and $R_3$, respectively.

Using the relations in (*), we describe some of the algebraic structures of cyclic codes over R.

(i) 

If $C_1=\langle u\rangle ,$$C_2=\langle {\beta }_{1}u+v\rangle$, and $C_3=\langle {\beta }_{2}u+v\rangle ,$ where ${\beta }_i$ are in $\mathsf{\Gamma }\left(3\right),$ then this means that the cyclic code, in $R+R_2+R_3$, is of the form

$$C\cong \langle u\rangle +\langle {\beta }_{1}u+v\rangle +\langle {\beta }_{2}u+v\rangle ,$$

and this is isomorphic, in ${\displaystyle \frac{R\left[x\right]}{\langle x^7-1\rangle }}$, to the cyclic code

$$C=\langle u{q}_2\left(x\right)q_3\left(x\right),({\beta }_{1}u+v)q_1\left(x\right)q_3\left(x\right),({\beta }_{2}u+v)q_1\left(x\right)q_2\left(x\right)\rangle .$$

(ii) 

If $C_1=\langle \alpha u+v\rangle ,$$C_2=\langle {\theta }_1,v\rangle$, and $C_3=\langle {\beta }_{1}u+v\rangle ,$ where $\alpha \in \Delta \left(4\right)$ and ${\beta }_1\in \mathsf{\Gamma }\left(3\right),$ then, in $R_1+R_2+R_3$, we have

$$C\cong \langle \alpha u+v\rangle +\langle u,v\rangle +\langle {\beta }_{1}u+v\rangle .$$

In ${\displaystyle \frac{R\left[x\right]}{\langle x^7-1\rangle }}$, this corresponds to

$$C=\langle (\alpha u+v)q_2\left(x\right)q_3\left(x\right),u{q}_1\left(x\right)q_3\left(x\right),v{q}_1\left(x\right)q_3\left(x\right),({\beta }_{1}u+v)q_1\left(x\right)q_2\left(x\right)\rangle .$$

(iii) 

Assume that $C_1=\langle v\rangle ,$$C_2=\langle u+v\rangle$, and $C_3=\langle {\beta }_{1}u+v\rangle ,$ where ${\beta }_1\in \mathsf{\Gamma }\left(3\right).$ In $R_1+R_2+R_3$, we obtain

$$C\cong \langle v\rangle +\langle u+v\rangle +\langle {\beta }_{1}u+v\rangle .$$

Then, as a correspondent in the ring ${\displaystyle \frac{R\left[x\right]}{\langle x^7-1\rangle }}$, the code C is of the form

$$C=\langle 2q_2\left(x\right)q_3\left(x\right),(u+v)q_1\left(x\right)q_3\left(x\right),({\beta }_{1}u+v)q_1\left(x\right)q_2\left(x\right)\rangle .$$

4. Matrices $\mathit{S}$ Associated with the MacWilliams Relation for the SWE

This section aims to construct the matrices S associated with the MacWilliams relation for the SWE in Equation (7). First, in light of Theorem 1, there are three different chains of ideals of length $=4$ in R, which are as follows:

$$\begin{array}{cc}\hfill & R=\langle 1\rangle \supset J=\langle u,v\rangle \supset \langle u+v\rangle \supset \mathrm{soc}\left(R\right)\supset 0;\hfill \\ \hfill & R=\langle 1\rangle \supset J=\langle u,v\rangle \supset \langle u\rangle \supset \mathrm{soc}\left(R\right)\supset 0;\hfill \\ \hfill & R=\langle 1\rangle \supset J=\langle u,v\rangle \supset \langle v\rangle \supset \mathrm{soc}\left(R\right)\supset 0.\hfill \end{array}$$

(9)

Furthermore, we have the orders of such ideals as

$$\begin{array}{ccc}\hfill \mid J\mid =(p^m-1)p^{3m},& \mid \langle u+v\rangle \mid =\mid \langle u\rangle \mid =\mid \langle v\rangle \mid =p^{2m},\hfill & \hfill \mid \mathrm{soc}\left(R\right)\mid =p^m.\end{array}$$

(10)

Examine the relation ∼ established in R such that $x\sim y$ if there exists $a\in U\left(R\right)$ such that $x=ay$. This indicates that the unit group of R acts on the set R via translation, thus establishing an equivalence connection. We represent the equivalence classes as ${\widehat{d}}_1,{\widehat{d}}_2,\cdots ,{\widehat{d}}_q$. Let R be defined as in Theorem 1; it may be readily demonstrated that

$$\begin{array}{ccc}{\widehat{d}}_1=\left\{0\right\},& {\widehat{d}}_2=U\left(R\right),& {\widehat{d}}_3=\langle u\rangle \setminus \mathrm{soc}\left(R\right),\\ & & \\ {\widehat{d}}_4=\langle v\rangle \setminus \mathrm{soc}\left(R\right),& {\widehat{d}}_5=\langle u+v\rangle \setminus \mathrm{soc}\left(R\right),& {\widehat{d}}_6=\mathrm{soc}\left(R\right)\setminus \left\{0\right\}.\end{array}$$

(11)

Suppose that, in what follows,

$$\begin{array}{cc}\hfill R_1={\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2-\gamma v^2,uv\rangle }},& R_2={\displaystyle \frac{{\mathbb{F}}_{2^m}[u,v]}{\langle u^2,v^2\rangle }}.\hfill \end{array}$$

(12)

In addition, we put

$$\begin{array}{cccccc}{d}_1=0,& d_2=1,& d_3=u,& d_4=v,& d_5=u+v& \mathrm{and}{d}_6=\alpha ,\end{array}$$

(13)

where

$$\alpha =\left\{\begin{array}{cc}{u}^2,\hfill & \mathrm{if}R=R_1,\hfill \\ uv,\hfill & \mathrm{if}R=R_2.\hfill \end{array}\right.$$

The determination of $\vartheta$ for any Frobenius ring is provided in [15]; next, we present the result.

Theorem 2

([15]). There exists $q\in {\mathbb{Z}}^{+}$ such that for $\vartheta :R\to \mathbb{C}$ and if $1\le i\le q,$ then

$$\vartheta \left(x\right)={\zeta }_1^{a_1}{\zeta }_2^{a_2}\cdots {\zeta }_q^{a_q},$$

(14)

is a generating character of $R,$ where ${\zeta }_i$ is a $p^i$-root of unity.

In

Table 1. $\vartheta$ for the ring R.

Ring	$(\mathit{R},+)$	$\mathit{\vartheta }$
${\displaystyle \frac{F_{p^m}[u,v]}{\langle u^2-\gamma v^2,uv\rangle }}$	${(\underset{\mathrm{m}-\mathrm{times}}{\underbrace{{\mathbb{Z}}_p\times \cdots \times {\mathbb{Z}}_p}})}^4$	$\vartheta (a_1+a_{2}u+a_{3}v+a_4{u}^2)={\prod }_{i=1}^4\vartheta \left(a_i\right)$
${\displaystyle \frac{F_{p^m}[u,v]}{\langle u^2-v^2,uv\rangle }}$	${(\underset{\mathrm{m}-\mathrm{times}}{\underbrace{{\mathbb{Z}}_p\times \cdots \times {\mathbb{Z}}_p}})}^4$	$\vartheta (a_1+a_{2}u+a_{3}v+a_4{u}^2)={\prod }_{i=1}^4\vartheta \left(a_i\right)$
${\displaystyle \frac{F_{2^m}[u,v]}{\langle u^2,v^2\rangle }}$	${(\underset{\mathrm{m}-\mathrm{times}}{\underbrace{{\mathbb{Z}}_p\times \cdots \times {\mathbb{Z}}_p}})}^4$	$\vartheta (a_1+a_{2}u+a_{3}v+a_{4}uv)={\prod }_{i=1}^4\vartheta \left(a_i\right)$

, we consider $\zeta$ to be a p-th root of unity, which leads us to the following formula:

$$\vartheta \left(a_i\right)={\zeta }^{(a_{1i}+a_{2i}+\cdots +a_{(m-1)i})},$$

(15)

where $a_i\in \underset{\mathrm{m}-\mathrm{times}}{\underbrace{{\mathbb{Z}}_p\times \cdots \times {\mathbb{Z}}_p}}$, which has a form of

$$a_i=a_{1i}{x}_0+a_{2i}{x}_1+\cdots +a_{(m-1)i}{x}_{m-1},$$

(16)

where $x_0,\dots ,x_{m-1}$ is a basis of ${\mathbb{F}}_{p^m}$ over ${\mathbb{Z}}_p.$

We define $S=\left(d_{ij}\right),$ where

$$d_{ij}=\sum _{b\in \widehat{d_j}}\vartheta \left(d_{i}b\right).$$

(17)

Moreover, we set

$${\Delta }_{ij}=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}{d}_i\in \mathrm{ann}\left({\widehat{d}}_j\right),\hfill \\ 0,\hfill & \mathrm{if}\mathrm{soc}\left(R\right)\cap d_i{\widehat{d}}_j=\varphi ,\hfill \\ {\displaystyle \frac{-1}{p^m-1}},\hfill & \mathrm{if}\mathrm{soc}\left(R\right)\cap d_i{\widehat{d}}_j\ne \varphi .\hfill \end{array}\right.$$

(18)

Proposition 1.

Let $R={\mathbb{F}}_{p^m}[u,v]$, of order $p^{4m}.$ Then, $d_{ij}$ can be computed as

$$d_{ij}={\Delta }_{ij}\mid \widehat{d_j}\mid .$$

(19)

Proof. 

Assume that $d_i\widehat{d_j}=0$. In this case, we have $d_i\in$ ann$\left(\widehat{d_j}\right)$, which means ${\Delta }_{ij}=1.$ Moreover, we obtain

$$d_{ij}=\sum _{b\in \widehat{d_j}}\vartheta \left(d_{i}b\right)=\sum _{b\in \widehat{d_j}}0=|\widehat{d_j}|={\Delta }_{ij}\left|\widehat{d_j}\right|.$$

For the other situation, assume $d_i\widehat{d_j}\ne \left\{0\right\}$. Let $\alpha \in d_i\widehat{d_j}$. First, we can show that all elements of $d_i\widehat{d_j}$ are non-zero because, if $0\in d_i\widehat{d_j},$ then there exists $x\in \widehat{d_j}$ with $d_{i}x=0.$ Suppose that $y\in \widehat{d_j};$ then, $x\sim y,$ and hence, $x=\beta y.$ Now, note that $0=d_{i}x=d_i\beta y,$ and thus, $d_{i}y=0.$ This shows that if $d_i\in \mathrm{ann}\left(x\right),$ where $x\in \widehat{d_j},$ then $d_i\in$ ann$\left(\widehat{d_j}\right).$ Since $\mathrm{soc}\left(R\right)=\langle \alpha \rangle$, it follows that $\alpha =\gamma y$, where $\gamma \in {\Delta }^{*}\left(m\right)$ and $y\in \widehat{d_j}$ represents the equivalence class $\widehat{d_j}$. Now, take $x\in d_i\widehat{d_j}$, meaning $x=d_i{y}^{\prime }$ for some $y^{\prime }$ in $\widehat{d_j}$. As y and $y^{\prime }$ are in $\widehat{d_j},$ it follows that $y\sim y^{\prime },$ and hence, there is a unit such that $\delta y=y^{\prime }.$ This means that $x=d_i\left(\delta y\right)$, and we also have $y={\gamma }^{-1}\alpha .$ Thus, $x=\delta {\gamma }^{-1}{d}_i\alpha ,$ and consequently, we can express $x=\Delta \alpha$ for some $\Delta \in {\Delta }^{*}\left(m\right)$. This indicates that all elements of $d_i\widehat{d_j}$ have the form $\gamma \alpha$, which suggests that the set $d_i\widehat{d_j}$ consists of copies of $\mathrm{soc}\left(R\right)$. Thus,

$$d_{ij}=N\sum _{\gamma \in {\Delta }^{*}\left(1\right)}{\zeta }^{\gamma }.$$

We know that in complex numbers,

$$\sum _{\gamma \in \Delta \left(1\right)}{\zeta }^{\gamma }=0.$$

(20)

Therefore, we obtain

$$\sum _{\gamma \in {\Delta }^{*}\left(1\right)}{\zeta }^{\gamma }=-1.$$

(21)

The positive N indicates the quantity of copies of soc$\left(R\right)$, which is exactly $N={\displaystyle \frac{1}{p^m-1}}\left|\widehat{d_j}\right|.$ Thus,

$$d_{ij}={\displaystyle \frac{-1}{p^m-1}}\left|\widehat{d_j}\right|.$$

(22)

This means that when soc$\left(R\right)\cap d_i\widehat{d_j}\ne \varphi ,$ it is also true that

$$d_{ij}={\Delta }_{ij}\left|\widehat{d_j}\right|.$$

(23)

Finally, we assume that soc$\left(R\right)\cap d_i\widehat{d_j}=\varphi .$ Let $y\in d_i\widehat{d_j};$ then, $y={\gamma }_0+{\gamma }_{1}u+{\gamma }_{2}v+\gamma \alpha ,$ where ${\gamma }_i,\gamma \in \Delta \left(m\right).$ Now, since $y\notin$ soc$\left(R\right),$y can be written as $x+\gamma \alpha$, where $x\ne 0.$ The proof can be approached in a similar manner to what is shown above. Since $\vartheta (x+\gamma \alpha )=\vartheta \left(x\right)·\vartheta \left(\gamma \alpha \right)$, we can fix x and write

$$\sum _{\gamma \in \Delta \left(m\right)}\vartheta \left(x\right)\vartheta \left(\gamma \alpha \right)=\vartheta \left(x\right)\sum _{\gamma \in \Delta \left(m\right)}\vartheta \left(\gamma \alpha \right).$$

Note that

$$\sum _{\gamma \in \Delta \left(m\right)}\vartheta \left(\gamma \alpha \right)=\sum _{\gamma \in \Delta \left(m\right)}\vartheta \left(\gamma \right).$$

Thus, we have

$$d_{ij}=\sum _{\gamma \in \Delta \left(m\right)}\sum _x\vartheta \left(x\right)\vartheta \left(\gamma \alpha \right).$$

Consequently, from Equation (20), we determine that ${\sum }_{\gamma \in \Delta \left(m\right)}\vartheta \left(\gamma \right)=0$. Therefore, we arrive at the conclusion that $d_{ij}=0$.

In conclusion, we have

$$d_{ij}={\Delta }_{ij}\mid \widehat{d_j}\mid .$$

(24)

□

We will now compute the matrices S of the symmetrized weight enumerators that correspond to the MacWilliams identities for R. The identities create an important link between weight enumerators and the dual of a code, which is crucial for coding theory. For the sake of clarity, we denote

$$ϵ=\left\{\begin{array}{cc}-p^m,\hfill & \mathrm{if}p=2,\hfill \\ p^m(p^m-1),\hfill & \mathrm{if}p\ne 2\hfill \end{array}\right.,$$

$${ϵ}_{\gamma }=\left\{\begin{array}{cc}-p^m,\hfill & \mathrm{if}\gamma \ne -1,\hfill \\ p^m(p^m-1),\hfill & \mathrm{if}\gamma =-1.\hfill \end{array}\right.$$

Theorem 3.

Assume $R={\mathbb{F}}_{p^m}[u,v]$ of order $p^{4m}.$ Then,

$$S\left(i\right)=\left(\begin{array}{cccc}1& (p^m-1)p^{3m}& (p^m-1)p^m& p^m-1\\ 1& 0& 0& -1\\ 1& 0& S\left(R_i\right)& p^m-1\\ 1& -p^{3m}& (p^m-1)p^m& p^m-1\end{array}\right),$$

(25)

where

$$\begin{array}{cc}\hfill S\left(R_1\right)=\left(\begin{array}{ccc}ϵ& -p^m& -p^m\\ -p^m& p^m(p^m-1)& -p^m\\ -p^m& -p^m& {ϵ}_{\gamma }\end{array}\right),& S\left(R_2\right)=\left(\begin{array}{ccc}-p^m& p^m(p^m-1)& -p^m\\ p^m(p^m-1)& -p^m& -p^m\\ -p^m& -p^m& -p^m\end{array}\right).\hfill \end{array}$$

Proof. 

The proof relies significantly on Proposition 1. This indicates that to determine the entries $s_{ij}$ of the matrix S, it suffices to analyze the subsets $a_i{\widehat{d}}_j$: specifically, we need to consider whether $0\in a_i{\widehat{d}}_j$, $\alpha \in a_i{\widehat{d}}_j$, or $\alpha \notin a_i{\widehat{d}}_j\ne \left\{0\right\}$.

First, let us examine the classes ${\widehat{d}}_j$ and their sizes $|{\widehat{d}}_j|$ as previously noted. By Equations (10) and (11),

$$\begin{array}{cccc}\hfill & |{\widehat{d}}_1|=1,\hfill & \hfill |{\widehat{d}}_2|=p^{3m}(p^m-1),& |{\widehat{d}}_3|=p^m(p^m-1),\hfill \\ & |{\widehat{d}}_4|=p^m(p^m-1),\hfill & \hfill |{\widehat{d}}_5|=p^m(p^m-1),& |{\widehat{d}}_6|=p^m-1.\hfill \end{array}$$

(26)

Additionally, assume $\mathrm{soc}\left(R\right)=\left(\alpha \right)$, with $d_1=0$, $d_2=1$, $d_3=u$, $d_4=v$, $d_5=u+v$, and $d_6=\alpha$. For all i, we have $d_i{\widehat{d}}_1=0$. Consequently, it follows that $d_{i1}=1$. Next, let us observe the following:

$$\left\{\begin{array}{cc}0\in d_1{\widehat{d}}_i,\hfill & \mathrm{if}1\le i\le 6,\hfill \\ 0\in d_6{\widehat{d}}_i,\hfill & \mathrm{if}3\le i\le 6,\hfill \\ 0\in d_i{\widehat{d}}_6,\hfill & \mathrm{if}2\le i\le 5.\hfill \end{array}\right.$$

Hence, the elements of such subsets are 0, and according to Proposition 1, we find that $d_{1i}=d_{6i}=d_{i6}=\left|{\widehat{d}}_i\right|$. On the other hand, it can be seen that $\alpha \notin d_2{\widehat{d}}_i\cup d_i{\widehat{d}}_2$ for $2\le i\le 5$. Therefore, for these values of i, we have $d_{2i}=0=d_{i2}$.

Now we treat the cases of $d_i{\widehat{d}}_j$ and $3\le i,j\le 5$, it is important to note that $\alpha$ is present in $d_5{\widehat{d}}_3$, $d_5{\widehat{d}}_4$, $d_3{\widehat{d}}_5$, and $d_4{\widehat{d}}_5$. Thus, based on the same lemma, we have

$$d_{ij}={\displaystyle \frac{-1}{p^m-1}}\left(p^m(p^m-1)\right)=-p^m.$$

Note that $d_{55}$ is related to $\gamma$. We recall that $d_5=\langle u+v\rangle$ and ${\widehat{d}}_5=\langle u+v\rangle \setminus \mathrm{soc}\left(R\right)$. In any case for R (a non-chain ring), we have $\alpha (\gamma +1)\in d_5{\widehat{d}}_5$. If $\gamma =-1$, then $0\in d_5{\widehat{d}}_5$, leading to $d_{55}=p^m(p^m-1)$. Conversely, if $\gamma \ne -1$, then $\alpha \in d_5{\widehat{d}}_5$, resulting in $s_{55}=-p$. In summary, we have

$$d_{55}={ϵ}_{\gamma },$$

where ${ϵ}_{\gamma }$ is as defined above. We observe that $d_{33}=d_{44}$ because $\langle u^2\rangle =\langle v^2\rangle$, except in the case of $R_2={\displaystyle \frac{{\mathbb{F}}_{2^m}[u,v]}{\langle u^2,v^2\rangle }},$ where $s_{33}=-2^m$ and $s_{44}=2^m$. Furthermore, $d_{34}=s_{43}$ for all rings, since $d_3{\widehat{d}}_4=d_4{\widehat{d}}_3$.

To conclude, we obtain two distinct $3\times 3$ submatrices of the following form:

$$\begin{array}{cc}\hfill S\left(R_1\right)=\left(\begin{array}{ccc}ϵ& -p^m& -p^m\\ -p^m& p^m(p^m-1)& -p^m\\ -p^m& -p^m& {ϵ}_{\gamma }\end{array}\right),& S\left(R_2\right)=\left(\begin{array}{ccc}-p^m& p^m(p^m-1)& -p^m\\ p^m(p^m-1)& -p^m& -p^m\\ -p^m& -p^m& {ϵ}_{\gamma }\end{array}\right).\hfill \end{array}$$

Thus, we derive $S\left(1\right)$ and $S\left(2\right)$ for $R_1$ and $R_2$, respectively, as intended. □

Example 3.

Suppose $R={\displaystyle \frac{{\mathbb{F}}_{2^3}[u,v]}{\langle u^2,v^2\rangle }}.$ The required matrix S is given by

$$S\left(2\right)=\left(\begin{array}{cccccc}1& 3584& 56& 56& 56& 7\\ 1& 0& 0& 0& 0& -1\\ 1& 0& -8& 56& 56& 7\\ 1& 0& 56& -8& -8& 7\\ 1& 0& -8& -8& -8& 7\\ 1& -512& 56& 56& 56& 7\end{array}\right).$$

(27)

5. Homogeneous Weight

Subsequently, we shall delineate the homogeneous weight of linear codes linked to Frobenius rings. Prior to presenting the precise definition of local Frobenius rings, we will outline the general notion of homogeneous weight for finite rings.

Definition 1.

A function $\omega :R\to \mathbb{R}$ is called a homogeneous weight when $\omega \left(0\right)=0$ and the following criteria hold:

H1. 

If $R{a}_1=R{a}_2,$ then $\omega \left(a_1\right)=\omega \left(a_2\right),$ where $a_1,a_2\in R.$

H2. 

There is $\delta \in \mathbb{R}$ such that

$$\sum _{r\in Ra}\omega \left(r\right)=\delta \left|Ra\right|.$$

(28)

The homogeneous weight is affected by the ideal structure of the ring R, and specific elements of the weight structure stem from the previously described requirements H1 and H2. We can inherently extend this concept to encompass linear codes over R by articulating it as follows: Suppose that C is a linear code of length $n,$ and $\mathbf{c}=(c_1,c_2,\cdots ,c_n)\in R^n.$ It follows that

$${\omega }_{hom}\left(\mathbf{c}\right)=\sum _{i=1}^n{\omega }_{hom}\left(c_i\right).$$

(29)

However, the paper deals with Frobenius ring $R,$ and thus, by [7], there is a complex-valued function $\varphi$ such that

$$\varphi \left(x\right)={\displaystyle \frac{1}{\left|U\right(R\left)\right|}}\sum _{a\in U\left(R\right)}\vartheta \left(ax\right).$$

(30)

By this function, we define the homogeneous weight $\omega$ for R as

$$\omega \left(x\right)=\delta [1-\varphi (x\left)\right],$$

(31)

where $\delta$ is a real number.

Theorem 4.

Let $x\in R.$ Then,

$$\omega \left(x\right)=\left\{\begin{array}{cc}0,\hfill & \mathit{if}x=0;\hfill \\ p^{3m},\hfill & \mathit{if}x\mathit{is}\mathrm{a}\mathit{non}-\mathit{zero}\mathit{element}\mathit{of}\mathit{soc}\left(R\right);\hfill \\ (p^m-1)p^{2m},\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

(32)

Proof. 

Since $\left|U\left(R\right)\right|=p^{4m}-p^{3m}=(p^m-1)p^{3m},$ it follows that, by Equation (30), we have

$$\omega \left(x\right)=(p^m-1)p^{m-2}-{\displaystyle \frac{1}{p^m}}\sum _{a\in U\left(R\right)}\vartheta \left(ax\right),$$

(33)

where $\delta$ is chosen to be

$$\delta =(p^m-1)p^{2m}.$$

The formula is valid for $x=0$. If $x\in \mathrm{soc}\left(R\right)$, indicating that $x=\alpha y$ is within the ideal $\langle \alpha \rangle$, then $ax$ is also included, and $\mathrm{Tr}\left(ax\right)=\mathrm{Tr}\left(ay\right)\alpha$. Furthermore, recall that $\zeta$ represents a primitive p-th root of unity, and y can be considered as belonging to ${\mathbb{F}}_{p^m}$ by associating $y\in {\mathbb{F}}_{p^m}$ with its Teichmüller representative. Additionally, $\mathrm{Tr}\left(ay\right)\alpha$ is dependent on a modulo J. For a given residue class modulo J, there exist $p^m$ potential values for a. We proceed to determine the number of solutions $a\in {\mathbb{F}}_{p^m}$ for the equation $\mathrm{Tr}\left(ay\right)=c\in {\mathbb{F}}_p$. It is noted that c has $p^{m-1}$ pre-images under Tr in ${\mathbb{F}}_{p^m}$, all of which are non-zero when c is non-zero; consequently, there are $p^{m-1}$ solutions to the equation when c is non-zero. For $c=0,$ there are $p^{m-1}$ pre-images under Tr; however, one of these is the zero element of ${\mathbb{F}}_{p^m},$ which we exclude, yielding $p^{m-1}-1$ solutions to the equation. The right side of the Formula (33) is expressed as

$$(p^m-1)p^{2m}-{\displaystyle \frac{1}{p^m}}{p}^{3m}\left[p^{m-1}-1+p^{m-1}\sum _{c\in F_p}{\zeta }^c\right].$$

Recalling that ${\sum }_{c\in F_p}{\zeta }^c=-1,$ the last expression simplifies to

$$(p^m-1)p^{2m}-p^{2m}\left[p^{m-1}-1-p^{m-1}\right]=p^{3m}$$

as intended. Let $x\notin \langle \alpha \rangle ,$ then, $x=d_{i}y$ for some unit y of R and some $d_i\ne \alpha ,$ a representative of a class Equation (13). As $y\in U\left(R\right),$ we can rewrite the sum as follows:

$$\sum _{a\in U}\vartheta \left(x\right)=\sum _{a\in U}\vartheta \left(d_{i}y\right)=\sum _{a\in U}{\zeta }^{\mathrm{Tr}\left(ay\right)}=\sum _{a\in U}{\zeta }^{\mathrm{Tr}\left(a\right)},$$

where

$$\zeta =\left\{\begin{array}{cc}\zeta ,\hfill & \mathrm{if}{d}_i\ne u+v,\hfill \\ {\zeta }^2,\hfill & \mathrm{if}{d}_i=u+v.\hfill \end{array}\right.$$

Thus, we need to show that

$$\sum _{a\in U}{\zeta }^{\mathrm{Tr}\left(a\right)}=0.$$

As x is in only one of the equivalence classes ${\widehat{d}}_i,$ where $i\ne 6$ or $i=0,$ it follows that if $x\in {\widehat{d}}_i,$ we obtain

$$\sum _{a\in U}{\zeta }^{\mathrm{Tr}\left(a\right)}=d_{i2}.$$

The second column of the matrix S implies that $d_{i2}=0.$ Thus,

$$\sum _{a\in U}{\zeta }^{\mathrm{Tr}\left(a\right)}=0.$$

Therefore, $\omega \left(x\right)=(p^m-1)p^{2m}.$ □

We shall now elucidate one of the principal findings of this paper. We characterize the homogeneous weight through the matrix S and employ the notion that the classes ${\widehat{d}}_i$ constitute a partition of $R.$ We utilize the second column of S, as stated in Theorem 3, which is

$$C_2=\left(\begin{array}{c}{p}^{3m}(p^m-1)\\ 0\\ 0\\ 0\\ 0\\ -p^{3m}\end{array}\right).$$

(34)

Theorem 5.

If $x\in {\widehat{d}}_i,$ then

$$\omega \left(x\right)=(p^m-1)p^{2m}\left[1-{\displaystyle \frac{1}{(p^m-1)p^{3m}}}{d}_{i2}\right].$$

(35)

Proof. 

Suppose that $x\in R.$ Since ∼ is an equivalence relation on $R,$ it follows that ${\widehat{d}}_i$s are partitions of $R.$ Thus, there is a unique i such that $x\in {\widehat{d}}_i.$ On the other hand, by Equations (30) and (31),

$$\omega \left(x\right)=(p^m-1)p^{2m}-{\displaystyle \frac{1}{p^m}}\sum _{a\in U\left(R\right)}\vartheta \left(ax\right),$$

(36)

which leads to

$$\omega \left(x\right)=(p^m-1)p^{2m}\left[1-{\displaystyle \frac{1}{p^{3m}(p^m-1)}}\sum _{a\in U\left(R\right)}\vartheta \left(ax\right)\right].$$

(37)

As $x\in {\widehat{d}}_i$ and ${\widehat{d}}_2=U\left(R\right),$ it follows that

$$d_{i2}=\sum _{a\in U\left(R\right)}\vartheta \left(ax\right).$$

Therefore,

$$\omega \left(x\right)=(p^m-1)p^{2m}\left[1-{\displaystyle \frac{1}{p^{3m}(p^m-1)}}{d}_{i2}\right].$$

(38)

Additionally, we have

$$C_2=\left(\begin{array}{c}{p}^{3m}(p^m-1)\\ 0\\ 0\\ 0\\ 0\\ -p^{3m}\end{array}\right).$$

(39)

If $x=0,$ then $0\in {\widehat{d}}_1=0,$ and hence, $d_{12}=p^{3m}(p^m-1).$ Hence, by (38), $\omega \left(x\right)=0.$ If $0\ne x\in$soc$\left(R\right),$ then $x\in {\widehat{d}}_6.$ Thus, $d_{26}=-p^{3m},$ and so

$$\begin{array}{cc}\hfill \omega \left(x\right)=& (p^m-1)p^{2m}\left[1-{\displaystyle \frac{1}{p^{3m}(p^m-1)}}(-p^{3m})\right]\hfill \\ \hfill =& (p^m-1)p^{2m}\left[1+{\displaystyle \frac{1}{p^m-1}}\right]\hfill \\ \hfill =& (p^m-1)p^{2m}\left[{\displaystyle \frac{p^m-1+1}{p^m-1}}\right]\hfill \\ \hfill =& p^{3m}.\hfill \end{array}$$

(40)

Finally, if $0\ne x\notin$soc$\left(R\right),$ then x is in only one of the following classes: ${\widehat{d}}_2,{\widehat{d}}_3,{\widehat{d}}_4$, and ${\widehat{d}}_5.$ In light of (34), we have $d_{22}=d_{23}=d_{24}=d_{25}=0.$ Thus, by Equation (38),

$$\omega \left(x\right)=p^{2m}(p^m-1).$$

This finishes the proof. □

Example 4.

Let $R={\displaystyle \frac{{\mathbb{F}}_{2^m}[u,v]}{\langle u^2,v^2\rangle }}.$ The homogeneous weight distribution of linear codes of length n over R is fully determined by the second column $C_2$ of $S\left(2\right)$:

$$C_2=\left(\begin{array}{c}{2}^{3m}(2^m-1)\\ 0\\ 0\\ 0\\ 0\\ -2^{3m}\end{array}\right).$$

(41)

In particular, we have

$$\omega \left(x\right)=\left\{\begin{array}{cc}0,\hfill & \mathit{if}x=0;\hfill \\ 2^{3m},\hfill & \mathit{if}0\ne x\in \mathit{soc}\left(R\right);\hfill \\ 2^{2m}(2^m-1),\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

(42)

In the case where $m=3,$ we have

$$C_2=\left(\begin{array}{c}3584\\ 0\\ 0\\ 0\\ 0\\ -512\end{array}\right).$$

(43)

The homogeneous weights over R are given by

$$\omega \left(x\right)=\left\{\begin{array}{cc}0,\hfill & \mathit{if}x=0,\hfill \\ 512,\hfill & \mathit{if}x\mathit{is}\mathrm{a}\mathit{non}-\mathit{zero}\mathit{element}\mathit{of}\langle uv\rangle ,\hfill \\ 448,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

(44)

6. Conclusions

Local rings are crucial in coding theory, positioning them as a key focus of research. Local Frobenius rings have received significant attention owing to their relevance in distance distributions, error correction, and the study of the SWE for linear codes and their duals. This article analyzes local rings ${\mathbb{F}}_{p^m}[u,v]$ with a length of four, focusing on those with rank $p^{4m}$. All such rings were successfully identified up to isomorphism, considering fixed invariants. Additionally, we classified each Frobenius ring based on the criteria outlined in Theorem 1. While Theorem 5 establishes a correlation between homogeneous distances in Frobenius rings and the matrices related to the MacWilliams identity for symmetrized weight enumerators. Future research should concentrate on more in-depth characterizations of homogeneous distances in Frobenius rings and explore their connections to the matrices related to the MacWilliams identity for symmetrized weight enumerators.