Complete Weight Enumerator of Torsions and Their Applications

Abstract

Let $R={\mathbb{F}}_2+w_1{\mathbb{F}}_2+w_2{\mathbb{F}}_2$, where $w_1^2=w_1$, $w_2^2=w_2$, $w_1{w}_2=w_2{w}_1=0$. Firstly, we explore the algebraic structure of the torsion codes of MacDonald codes of types $\alpha$ and $\beta$ over R. Secondly, the complete weight enumerator of ${\mathcal{S}}_{k,h}^{\alpha }$ and ${\mathcal{S}}_{k,h}^{\beta }$ are given. Finally, as applications, we construct the secret sharing schemes and systematic authentication codes.

1. Introduction

Linear codes with a few weights are extensively utilized in applications such as authentication codes [1], secret sharing schemes [2], association schemes [3], and constant composition codes [4]. Furthermore, these codes are closely linked to various mathematical constructs including strongly regular graphs, partial geometries, and projective point sets. Consequently, the development of linear codes with a few weights has become a vibrant area of interest within coding theory.

MacDonald codes fall into the category of linear codes distinguished by having two non-zero weights. The binary MacDonald codes were first introduced by MacDonald [5], while MacDonald codes over finite fields were explored in ref. [6]. Colbourn et al. in ref. [7] studied the MacDonald codes over ${\mathbb{Z}}_4$. Furthermore, Dertli et al. in [8] delved into the research of MacDonald codes over ${\mathbb{F}}_2+v{\mathbb{F}}_2$ with $v^2=v$, contributing to the comprehensive exploration of MacDonald codes across various mathematical structures. Afterwards, the research on the secret sharing scheme founded on the torsion codes of MacDonald codes over finite rings has drawn considerable attention [9,10,11,12]. Recently, secret sharing schemes using linear codes with a few weights were given in refs. [13,14,15,16]. Melakhessou et al. in ref. [17] investigated the DNA multi-secret sharing schemes.

As we know, the complete weight enumerator has significant applications in authentication codes and constant composition codes. Over the years, the exploration of weight distributions of linear codes over finite fields and finite rings has received a great deal of attention [18,19,20,21,22,23,24,25]. Recently, the enumerator of weight for the torsion codes of MacDonald codes over ${\mathbb{F}}_p+v{\mathbb{F}}_p$ [11] and ${\mathbb{F}}_p+v{\mathbb{F}}_p+v^2{\mathbb{F}}_p$ [26] were studied. Moreover, the ring ${\mathbb{F}}_2+w_1{\mathbb{F}}_2+w_2{\mathbb{F}}_2$ outperforms ${\mathbb{Z}}_4$ and ${\mathbb{F}}_2+v{\mathbb{F}}_2$ in terms of flexibility, computational efficiency, error correction capabilities, and applicability. Inspired by the above work, we take into account studying MacDonald codes and their torsion codes over ${\mathbb{F}}_2+w_1{\mathbb{F}}_2+w_2{\mathbb{F}}_2$ and the complete weight enumerator of two types of torsion codes, where $w_1^2=w_1,w_2^2=w_2,w_1{w}_2=w_2{w}_1$. As applications, the secret sharing schemes and systematic authentication codes were constructed.

The paper is outlined as below. In Section 2, the algebraic structure of the torsion codes of the MacDonald codes of types $\alpha$ and $\beta$ over R is given. In Section 3, we obtain the complete weight enumerator of ${\mathcal{S}}_{k,h}^{\alpha }$, and ${\mathcal{S}}_{k,h}^{\beta }$. In Section 4, as applications, we construct the secret sharing schemes and systematic authentication codes. Section 5 makes a summary of the paper.

2. MacDonald Codes over ${\mathbb{F}}_{\mathbf{2}}\mathbf{+}{\mathit{w}}_{\mathbf{1}}{\mathbb{F}}_{\mathbf{2}}\mathbf{+}{\mathit{w}}_{\mathbf{2}}{\mathbb{F}}_{\mathbf{2}}$

Set $R={\mathbb{F}}_2+w_1{\mathbb{F}}_2+w_2{\mathbb{F}}_2$ as a finite commutative non-chain ring with characteristic 2, where $w_1^2=w_1$, $w_2^2=w_2$, $w_1{w}_2=w_2{w}_1=0$. Based on the Chinese Remainder Theorem, R can be expressed as $(1-w_1-w_2)R\oplus w_{1}R\oplus w_{2}R$. Furthermore, $R=(1-w_1-w_2)R\oplus w_{1}R\oplus w_{2}R=(1-w_1-w_2){\mathbb{F}}_2\oplus w_1{\mathbb{F}}_2\oplus w_2{\mathbb{F}}_2$.

If $\varnothing \ne \mathcal{C}\subseteq R^n$ and $\mathcal{C}$ is an R-submodule of $R^n$, then $\mathcal{C}$ is a linear code of length n over R. Set $\mathcal{C}$ as a linear code and

$$\begin{array}{cc}\hfill & {\mathcal{C}}_0=\{\mathit{d}\in {\mathbb{F}}_2^n|\exists \mathit{e},\mathit{f}\in {\mathbb{F}}_2^n,(1-w_1-w_2)\mathit{d}+w_1\mathit{e}+w_2\mathit{f}\in \mathcal{C}\},\hfill \\ \hfill & {\mathcal{C}}_1=\{\mathit{e}\in {\mathbb{F}}_2^n|\exists \mathit{d},\mathit{f}\in {\mathbb{F}}_2^n,(1-w_1-w_2)\mathit{d}+w_1\mathit{e}+w_2\mathit{f}\in \mathcal{C}\},\hfill \\ \hfill & {\mathcal{C}}_2=\{\mathit{f}\in {\mathbb{F}}_2^n|\exists \mathit{d},\mathit{e}\in {\mathbb{F}}_2^n,(1-w_1-w_2)\mathit{d}+w_1\mathit{e}+w_2\mathit{f}\in \mathcal{C}\}.\hfill \end{array}$$

So ${\mathcal{C}}_0$, ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$ are all linear codes of length n over ${\mathbb{F}}_2$. The torsion codes of $\mathcal{C}$ are taken as ${\mathcal{C}}_0$, ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$.

Set $S_k^{\alpha }$ as a type $\alpha$ simplex code over R. The generator matrix $G_k^{\alpha }$ of $S_k^{\alpha }$ is built inductively. Set $G_k^{\alpha }$ as a $k\times 2^{3k}$ matrix and $G_1^{\alpha }=[01w_1{w}_{2}1+w_{1}1+w_2{w}_1+w_{2}1+w_1+w_2]$. Then, $G_k$ is built inductively as below

$$G_k^{\alpha }=\left[\begin{array}{cccccccc}0& 1& w_1& w_2& 1+w_1& 1+w_2& w_1+w_2& 1+w_1+w_2\\ G_{k-1}^{\alpha }& G_{k-1}^{\alpha }& G_{k-1}^{\alpha }& G_{k-1}^{\alpha }& G_{k-1}^{\alpha }& G_{k-1}^{\alpha }& G_{k-1}^{\alpha }& G_{k-1}^{\alpha }\end{array}\right].$$

Set $G(S_k)$, columns composed of all non-zero 2-ary k-tuples, as the generator matrix of ${[n,k]}_2$ simplex code $S_k$. The extended simplex code ${\widehat{S}}_k$ is produced by

$$G({\widehat{S}}_k)=\left[\begin{array}{cc}0& G(S_k)\end{array}\right],$$

where

$$G({\widehat{S}}_1)=\left[\begin{array}{cc}0& 1\end{array}\right],$$

and

$$G({\widehat{S}}_k)=\left[\begin{array}{cc}0\cdots 0& 1\cdots 1\\ G({\widehat{S}}_{k-1})& G({\widehat{S}}_{k-1})\end{array}\right].$$

Proposition 1. 

The torsion code ${\mathcal{C}}_0$ (or ${\mathcal{C}}_1$, ${\mathcal{C}}_2$) of $S_k^{\alpha }$ is equivalent under permutation to $2^{2k}$ replicas of ${\widehat{S}}_k$.

Proof. 

We demonstrate the ${\mathcal{C}}_0$ case by generalization on k. The generator matrix of ${\mathcal{C}}_0$ is derived by using 1 to replace $1-w_1-w_2$ in the matrix rows $(1-w_1-w_2)G_k^{\alpha }$. For $k=2$, it can be easily verified. If $(1-w_1-w_2)G_{k-1}^{\alpha }$ is equivalent under permutation to $2^{2(k-1)}$ replicas of $(1-w_1-w_2)G({\widehat{S}}_{k-1})$, then we have

$$(1-w_1-w_2)G_k^{\alpha }=\left[\begin{array}{cccc}D& D& D& D\end{array}\right],$$

where

$$D={\left[\begin{array}{cc}{G}_0& G_1\end{array}\right]}_{k\times 2^{3k-2}},$$

and

$$G_r=\left[\begin{array}{ccc}(1-w_1-w_2)r& \cdots & (1-w_1-w_2)r\\ (1-w_1-w_2)G({\widehat{S}}_{k-1})& \cdots & (1-w_1-w_2)G({\widehat{S}}_{k-1})\end{array}\right],r=0,1.$$

According to the hypothesis, we obtain that the size of $G_r$ is $k\times 2^{3k-3}$. Rearranging the columns, we achieve the result. □

The generator matrix $G_k^{\alpha }$ of simplex code $S_k^{\alpha }$ is able to be used to construct the MacDonald codes of type $\alpha$ over R. For $1\le h\le k-1$, define $G_{k,h}^{\alpha }$ as the matrix derived from $G_k^{\alpha }$ by removing columns corresponding to the columns of $G_h^{\alpha }$, i.e.,

$$G_{k,h}^{\alpha }=\left[\begin{array}{c}\hfill G_k^{\alpha }\setminus {\displaystyle \frac{\mathbf{0}}{G_h^{\alpha }}}\end{array}\right],$$

where $[E\setminus F]$ represents the matrix derived by removing matrix F from matrix E, and $\mathbf{0}$ is an $(k-h)\times 2^{3h}$ matrix.

Definition 1. 

The code ${\mathcal{C}}_{k,h}^{\alpha }$ produced by $G_{k,h}^{\alpha }$ is named as a type α MacDonald code.

Obviously, the code ${\mathcal{C}}_{k,h}^{\alpha }$ is a linear code of length $2^{3k}-2^{3h}$ over R. Denote ${\mathcal{S}}_{k,h}^{\alpha }$ as the torsion code of ${\mathcal{C}}_{k,h}^{\alpha }$. The generator matrix of ${\mathcal{S}}_{k,h}^{\alpha }$ is derived using 1 to replace $w_1$ in the matrix $w_1{G}_{k,h}^{\alpha }$. Likewise, we can obtain other torsion codes of ${\mathcal{C}}_{k,h}^{\alpha }$ by using 1 to replace $w_2$ or $1-w_1-w_2$ in $w_2{G}_{k,h}^{\alpha }$ or $(1-w_1-w_2)G_{k,h}^{\alpha }$. By Proposition 1, we are aware that the three torsion codes are mutually equivalent. Therefore, we concentrate on ${\mathcal{S}}_{k,h}^{\alpha }$.

The length of the simplex code of type $\alpha$ is substantial and grows rapidly. We can eliminate certain columns from $S_k^{\alpha }$. A type $\beta$ simplex code $S_k^{\beta }$ is obtained from $G_k^{\alpha }$ by removing some columns.

Set ${\lambda }_k$ as an $k\times (2^{3k}-2^{2k})$ matrix. Make ${\lambda }_1=\left[\begin{array}{cccc}1& 1+w_2& w_1+w_2& w_1\end{array}\right]$ and

$${\lambda }_2=\left[\begin{array}{cccccccc}0& 1& 1+w_2& w_1+w_2& w_1& w_2& 1+w_1& 1+w_1+w_2\\ {\lambda }_1& G_1^{\alpha }& G_1^{\alpha }& G_1^{\alpha }& G_1^{\alpha }& {\lambda }_1& {\lambda }_1& {\lambda }_1\end{array}\right],$$

then ${\lambda }_k$ is built inductively in the following manner

$${\lambda }_k=\left[\begin{array}{cccccccc}0& 1& 1+w_2& w_1+w_2& w_1& w_2& 1+w_1& 1+w_1+w_2\\ {\lambda }_{k-1}& G_{k-1}^{\alpha }& G_{k-1}^{\alpha }& G_{k-1}^{\alpha }& G_{k-1}^{\alpha }& {\lambda }_{k-1}& {\lambda }_{k-1}& {\lambda }_{k-1}\end{array}\right].$$

Set ${\delta }_k$ as an $k\times (2^{3k}-2^{2k})$ matrix. Make ${\delta }_1=\left[\begin{array}{cccc}1& 1+w_1& w_1+w_2& w_2\end{array}\right]$ and

$${\delta }_2=\left[\begin{array}{cccccccc}0& 1& 1+w_1& w_1+w_2& w_2& w_1& 1+w_2& 1+w_2+w_1\\ {\delta }_1& G_1^{\alpha }& G_1^{\alpha }& G_1^{\alpha }& G_1^{\alpha }& {\delta }_1& {\delta }_1& {\delta }_1\end{array}\right],$$

then ${\delta }_k$ is built inductively in the following manner

$${\delta }_k=\left[\begin{array}{cccccccc}0& 1& 1+w_1& w_1+w_2& w_2& w_1& 1+w_2& 1+w_2+w_1\\ {\delta }_{k-1}& G_{k-1}^{\alpha }& G_{k-1}^{\alpha }& G_{k-1}^{\alpha }& G_{k-1}^{\alpha }& {\delta }_{k-1}& {\delta }_{k-1}& {\delta }_{k-1}\end{array}\right].$$

Set ${\sigma }_k$ as an $k\times (2^{3k}-2^{2k})$ matrix. Make ${\sigma }_1=\left[\begin{array}{cccc}1& 1+w_2& 1+w_1& 1+w_1+w_2\end{array}\right]$ and

$${\sigma }_2=\left[\begin{array}{cccccccc}0& 1& 1+w_2& 1+w_1& 1+w_1+w_2& w_1& w_2& w_1+w_2\\ {\sigma }_1& G_1^{\alpha }& G_1^{\alpha }& G_1^{\alpha }& G_1^{\alpha }& {\sigma }_1& {\sigma }_1& {\sigma }_1\end{array}\right],$$

then ${\sigma }_k$ is built inductively in the following manner

$${\sigma }_k=\left[\begin{array}{cccccccc}0& 1& 1+w_2& 1+w_1& 1+w_1+w_2& w_1& w_2& w_1+w_2\\ {\sigma }_{k-1}& G_{k-1}^{\alpha }& G_{k-1}^{\alpha }& G_{k-1}^{\alpha }& G_{k-1}^{\alpha }& {\sigma }_{k-1}& {\sigma }_{k-1}& {\sigma }_{k-1}\end{array}\right].$$

Set ${\varrho }_k$ as an $k\times 2^k{(2^k-1)}^2$ matrix. Make ${\varrho }_1=\left[\begin{array}{cc}1& 1+w_1\end{array}\right]$ and

$${\varrho }_2=\left[\begin{array}{cccccccc}0& 1& 1+w_1& w_2& 1+w_2& w_1& 1+w_1& w_1\\ {\varrho }_1& G_1^{\alpha }& G_1^{\alpha }& {\delta }_1& {\delta }_1& {\sigma }_1& {\sigma }_1& {\varrho }_1\end{array}\right],$$

then ${\varrho }_k$ is built inductively in the following manner

$${\varrho }_k=\left[\begin{array}{cccccccc}0& 1& 1+w_1& w_2& 1+w_2& w_1& 1+w_1& w_1\\ {\varrho }_{k-1}& G_{k-1}^{\alpha }& G_{k-1}^{\alpha }& {\delta }_{k-1}& {\delta }_{k-1}& {\sigma }_{k-1}& {\sigma }_{k-1}& {\varrho }_{k-1}\end{array}\right].$$

Set ${\tau }_k$ as an $k\times 2^k{(2^k-1)}^2$ matrix. Make ${\tau }_1=\left[\begin{array}{cc}1& 1+w_2\end{array}\right]$ and

$${\tau }_2=\left[\begin{array}{cccccccc}0& 1& 1+w_2& w_2& 1+w_1& w_1& 1+w_2& w_2\\ {\tau }_1& G_1^{\alpha }& G_1^{\alpha }& {\lambda }_1& {\lambda }_1& {\sigma }_1& {\sigma }_1& {\tau }_1\end{array}\right],$$

then ${\tau }_k$ is built inductively in the following manner

$${\tau }_k=\left[\begin{array}{cccccccc}0& 1& 1+w_2& w_2& 1+w_1& w_1& 1+w_2& w_2\\ {\tau }_{k-1}& G_{k-1}^{\alpha }& G_{k-1}^{\alpha }& {\lambda }_{k-1}& {\lambda }_{k-1}& {\sigma }_{k-1}& {\sigma }_{k-1}& {\tau }_{k-1}\end{array}\right].$$

Set ${\upsilon }_k$ as an $k\times 2^k{(2^k-1)}^2$ matrix. Make ${\upsilon }_1=\left[\begin{array}{cc}1& w_1+w_2\end{array}\right]$ and

$${\nu }_2=\left[\begin{array}{cccccccc}0& 1& w_1+w_2& 1+w_1& w_2& 1+w_2& w_1& 1+w_1+w_2\\ {\nu }_1& G_1^{\alpha }& G_1^{\alpha }& {\lambda }_1& {\lambda }_1& {\delta }_1& {\delta }_1& {\nu }_1\end{array}\right],$$

then ${\nu }_k$ is built inductively in the following manner

$${\nu }_k=\left[\begin{array}{cccccccc}0& 1& w_1+w_2& 1+w_1& w_2& 1+w_2& w_1& 1+w_1+w_2\\ {\nu }_{k-1}& G_{k-1}^{\alpha }& G_{k-1}^{\alpha }& {\lambda }_{k-1}& {\lambda }_{k-1}& {\delta }_{k-1}& {\delta }_{k-1}& {\nu }_{k-1}\end{array}\right].$$

Set $G_k^{\beta }(k\ge 2)$ as an $k\times {(2^k-1)}^3$ generator matrix of $S_k^{\beta }$. Make

$$G_2^{\beta }=\left[\begin{array}{cccccccc}1& 0& 1+w_1& 1+w_2& w_1+w_2& w_1& w_2& 1+w_1+w_2\\ G_1^{\alpha }& 1& {\lambda }_1& {\delta }_1& {\sigma }_1& {\varrho }_1& {\tau }_1& {\nu }_1\end{array}\right],$$

then $G_k^{\beta }$ is built inductively in the following manner

$$G_k^{\beta }=\left[\begin{array}{cccccccc}1& 0& 1+w_1& 1+w_2& w_1+w_2& w_1& w_2& 1+w_1+w_2\\ G_{k-1}^{\alpha }& G_{k-1}^{\beta }& {\lambda }_{k-1}& {\delta }_{k-1}& {\sigma }_{k-1}& {\varrho }_{k-1}& {\tau }_{k-1}& {\nu }_{k-1}\end{array}\right].$$

Proposition 2. 

The torsion codes ${\mathcal{C}}_0$, ${\mathcal{C}}_1$, and ${\mathcal{C}}_2$ of $S_k^{\beta }$ are mutually equivalent.

Proof. 

The proof process parallels that of Proposition 1. □

Resembling the construction method of type $\alpha$ MacDonald codes, type $\beta$ MacDonald codes can be constructed. For $1\le h\le k-1$, define $G_{k,h}^{\beta }$ as the matrix derived from $G_k^{\beta }$ by removing the columns corresponding to the columns of $G_h^{\beta }$, i.e.,

$$G_{k,h}^{\beta }=\left(\begin{array}{c}\hfill G_k^{\beta }\setminus {\displaystyle \frac{\mathbf{0}}{G_h^{\beta }}}\end{array}\right),$$

where $[E\setminus F]$ represents the matrix derived by removing matrix F from matrix E, and $\mathbf{0}$ is an $(k-h)\times {(2^h-1)}^3$ matrix.

Definition 2. 

The code ${\mathcal{C}}_{k,h}^{\beta }$ produced by $G_{k,h}^{\beta }$ is named as a type β MacDonald code.

Denote ${\mathcal{S}}_{k,h}^{\beta }$ as the torsion code of ${\mathcal{C}}_{k,h}^{\beta }$ and ${\mathcal{C}}_{k,h}^{\beta }$ is a linear code of length ${(2^k-1)}^3-{(2^h-1)}^3$ over R. The generator matrix of ${\mathcal{S}}_{k,h}^{\beta }$ is derived using 1 to replace $w_1$ in the matrix $w_1{G}_{k,h}^{\beta }$. Likewise, we can obtain other torsion codes of ${\mathcal{C}}_{k,h}^{\beta }$ by using 1 to replace $w_2$ or $1-w_1-w_2$ in $w_2{G}_{k,h}^{\beta }$ and $(1-w_1-w_2)G_{k,h}^{\beta }$. According to Proposition 2, we are aware that the three torsion codes are mutually equivalent. Therefore, we concentrate on ${\mathcal{S}}_{k,h}^{\beta }$.

3. Complete Weight Enumerator of ${\mathcal{S}}_{\mathit{k}\mathbf{,}\mathit{h}}^{\mathit{\alpha }}$ and ${\mathcal{S}}_{\mathit{k}\mathbf{,}\mathit{h}}^{\mathit{\beta }}$

In this section, we are committed to exploring the complete weight enumerator of torsion codes ${\mathcal{S}}_{k,h}^{\alpha }$ and ${\mathcal{S}}_{k,h}^{\beta }$. First, we give the meaning of the complete weight enumerator of codes.

Denote $Y_0,Y_1$ as 2 indeterminates. Suppose $\mathcal{C}$ is a code of length n over ${\mathbb{F}}_2$. For any $e=(e_0,e_1,\dots ,e_{n-1})\in \mathcal{C}$, the weight of e at $l\in {\mathbb{F}}_2$ is regarded as

$${\phi }_l(e)=|\{i|e_i=l,i=0,1,\dots ,n-1\}|.$$

The composition of e is regarded as $\mathrm{comp}(e)=Y_0^{{\phi }_0(e)}{Y}_0^{{\phi }_1(e)}$. The complete weight enumerator of $\mathcal{C}$ is regarded as

$$W_{\mathcal{C}}(Y_0,Y_1)=\sum _{e\in \mathcal{C}}{Y}_0^{{\phi }_0(e)}{Y}_0^{{\phi }_1(e)}.$$

In the interest of brevity, we employ $|G|$ to stand for the column count of the matrix G. Specifically, let $|G_0^{\alpha }|=1$, $|{\lambda }_0|=1$, $|{\delta }_0|=1$, $|{\sigma }_0|=1$, $|{\varrho }_0|=1$, $|{\tau }_0|=1$, $|{\nu }_0|=1$. We employ ${(G)}_r$ to stand for the r-th row of the matrix G. Let $G_{k,S}^{\alpha }$ denote the matrix derived from $w_1{G}_k^{\alpha }$ by substituting 1 for $w_1$, and similarly, let $G_{k,h,S}^{\alpha }$ represent the matrix obtained from $w_1{G}_{k,h}^{\alpha }$ by substituting 1 for $w_1$, where $G_{k,h,S}^{\alpha }$ serves as the generator matrix for the torsion code ${\mathcal{S}}_{k,h}^{\alpha }$. Subsequently, we aim to establish the compositions of the r-th rows, denoted as ${\left(G_{k,S}^{\alpha }\right)}_r$ and ${\left(G_{k,h,S}^{\alpha }\right)}_r$, across all indices r from 1 to k.

Lemma 1. 

For $r=1,\dots ,k$, the following is true:

$$comp\left({\left(G_{k,S}^{\alpha }\right)}_r\right)=Y_0^{2^{3k-1}}{Y}_1^{2^{3k-1}}.$$

Proof. 

Through the process of constructing $G_k^{\alpha }$, it follows that ${\left(G_{k,S}^{\alpha }\right)}_r={\left({\left(G_{k-r+1,S}^{\alpha }\right)}_1\right)}^{2^{3(r-1)}}$. Hence, $\mathrm{comp}\left({\left(G_{k,S}^{\alpha }\right)}_r\right)={\left(\mathrm{comp}\left({\left(G_{k-r+1,S}^{\alpha }\right)}_1\right)\right)}^{2^{3(r-1)}}$. Due to $\left(\mathrm{comp}\left({\left(G_{k-r+1,S}^{\alpha }\right)}_1\right)\right)=Y_0^{4|G_{k-r}^{\alpha }|}{Y}_1^{4|G_{k-r}^{\alpha }|}$, we have

$${\left(\mathrm{comp}\left({\left(G_{k-r+1,S}^{\alpha }\right)}_1\right)\right)}^{2^{3(r-1)}}=Y_0^{2^{3r-1}|G_{k-r}^{\alpha }|}{Y}_1^{2^{3r-1}|G_{k-r}^{\alpha }|}.$$

Moreover, $|G_{k-r}^{\alpha }|=2^{3(k-r)}$. Consequently,

$${\left(\mathrm{comp}\left({\left(G_{k-r+1,S}^{\alpha }\right)}_1\right)\right)}^{2^{3(r-1)}}=Y_0^{2^{3k-1}}{Y}_1^{2^{3k-1}}.$$

□

The following conclusion can be derived directly from Lemma 1.

Proposition 3. 

1. 

Provided that $1\le r\le k-h$, $1\le h\le k-1$, we have

$$\mathrm{comp}\left({\left(G_{k,h,S}^{\alpha }\right)}_r\right)=Y_0^{2^{3k-1}-2^{3h}}{Y}_1^{2^{3k-1}}.$$

2. 

Provided that $k-h+1\le r\le k$, $1\le h\le k-1$, we have

$$\mathrm{comp}\left({\left(G_{k,h,S}^{\alpha }\right)}_r\right)=Y_0^{2^{3k-1}-2^{3h-1}}{Y}_1^{2^{3k-1}-2^{3h-1}}.$$

Proposition 4. 

For $1\le r<s\le k$, we have

$$\mathrm{comp}\left({(G_{k,h,S}^{\alpha })}_r+{(G_{k,h,S}^{\alpha })}_s\right)=\mathrm{comp}\left({(G_{k,h,S}^{\alpha })}_s\right).$$

Proof. 

Let

$$A_0={\left[\begin{array}{cccc}0& 0& \cdots & 0\\ {(G_{k-s+1,S}^{\alpha })}_1& {(G_{k-s+1,S}^{\alpha })}_1& \cdots & {(G_{k-s+1,S}^{\alpha })}_1\end{array}\right]}_{2\times 2^{3k-3r}},$$

and

$$A_1={\left[\begin{array}{cccc}1& 1& \cdots & 1\\ {(G_{k-s+1,S}^{\alpha })}_1& {(G_{k-s+1,S}^{\alpha })}_1& \cdots & {(G_{k-s+1,S}^{\alpha })}_1\end{array}\right]}_{2\times 2^{3k-3r}},$$

where the number of ${(G_{k-s+1,S}^{\alpha })}_1$ in $A_0$ and $A_1$ is $2^{3(s-r-1)}$. Let

$$A={\left[\begin{array}{cccccccc}{A}_0& A_1& A_0& A_0& A_0& A_1& A_1& A_1\end{array}\right]}_{2\times 2^{3(k-r+1)}}.$$

Moving forward, we will explore the constructions of ${(G_{k,h,S}^{\alpha })}_r$ and ${(G_{k,h,S}^{\alpha })}_s$ in three different scenarios.

Case I: $1\le r<s\le k-h$, $1\le h\le k-1$. Let

$$A_0^{†}={\left[\begin{array}{cccc}0& 0& \cdots & 0\\ {(G_{k-s+1,S}^{\alpha })}_1\setminus 0& {(G_{k-s+1,S}^{\alpha })}_1& \cdots & {(G_{k-s+1,S}^{\alpha })}_1\end{array}\right]}_{2\times (2^{3k-3r}-2^{3h})},$$

where the number of ${(G_{k-s+1,S}^{\alpha })}_1$ in $A_0^{†}$ is $2^{3(s-r-1)}-1$. Let

$$A^{†}={\left[\begin{array}{cccccccc}{A}_0^{†}& A_1& A_0& A_0& A_0& A_1& A_1& A_1\end{array}\right]}_{2\times (2^{3(k-r+1)}-2^{3h})}.$$

Then, we have

$$\left[\begin{array}{c}{(G_{k-s+1,S}^{\alpha })}_r\\ {(G_{k-s+1,S}^{\alpha })}_s\end{array}\right]={\left[\begin{array}{ccccc}{A}^{†}& A& A& \cdots & A\end{array}\right]}_{2\times (2^{3k}-2^{3h})},$$

where the number of A is $2^{3(r-1)}-1$.

Case II: $1\le r\le k-h$, $k-h+1\le s\le k$ and $1\le h\le k-1$. Let

$$A_0^{‡}={\left[\begin{array}{cccc}0& 0& \cdots & 0\\ {(G_{k-s+1,S}^{\alpha })}_1& {(G_{k-s+1,S}^{\alpha })}_1& \cdots & {(G_{k-s+1,S}^{\alpha })}_1\end{array}\right]}_{2\times (2^{3k-3r}-2^{3h})},$$

where the number of ${(G_{k-s+1,S}^{\alpha })}_1$ in $A_0^{‡}$ is $2^{3(s-r-1)}-2^{3(h-k+s-1)}$. Let

$$A^{‡}={\left[\begin{array}{cccccccc}{A}_0^{‡}& A_1& A_0& A_0& A_0& A_1& A_1& A_1\end{array}\right]}_{2\times (2^{3(k-r+1)}-2^{3h})}.$$

Then, we have

$$\left[\begin{array}{c}{(G_{k-s+1,S}^{\alpha })}_r\\ {(G_{k-s+1,S}^{\alpha })}_s\end{array}\right]={\left[\begin{array}{ccccc}{A}^{‡}& A& A& \cdots & A\end{array}\right]}_{2\times (2^{3k}-2^{3h})},$$

where the number of A is $2^{3(r-1)}-1$.

Case III: $k-h+1\le r<s\le k$, $1\le h\le k-1$. We have

$$\left[\begin{array}{c}{(G_{k-s+1,S}^{\alpha })}_r\\ {(G_{k-s+1,S}^{\alpha })}_s\end{array}\right]={\left[\begin{array}{ccccc}A& A& A& \cdots & A\end{array}\right]}_{2\times (2^{3k}-2^{3h})},$$

where the number of A is $2^{3(r-1)}-2^{h-k+r-1}$.

Based on the constructions of ${(G_{k-s+1,S}^{\alpha })}_r$ and ${(G_{k-s+1,S}^{\alpha })}_s$, we can ascertain the composition of ${(G_{k-s+1,S}^{\alpha })}_r+{(G_{k-s+1,S}^{\alpha })}_s$ for any $1\le r<s\le k$.

Since 0 and 1 appear at the same time in ${(G_{k-s+1,S}^{\alpha })}_1$, then we have $\mathrm{comp}({(A_0)}_1+{(A_0)}_2)=\mathrm{comp}({(A_0)}_2)$ and $\mathrm{comp}({(A_1)}_1+{(A_1)}_2)=\mathrm{comp}({(A_1)}_2)$. Consequently,

$$\begin{array}{cc}\hfill & \mathrm{comp}({(A)}_1+{(A)}_2)\hfill \\ \hfill =& {(\mathrm{comp}({(A_0)}_1+{(A_0)}_2))}^4{(\mathrm{comp}({(A_1)}_1+{(A_1)}_2))}^4\hfill \\ \hfill =& {(\mathrm{comp}({(A_0)}_2))}^4{(\mathrm{comp}({(A_1)}_2))}^4\hfill \\ \hfill =& \mathrm{comp}({(A)}_2).\hfill \end{array}$$

Concerning $A_0^{†}$, it is the case that

$$\begin{array}{cc}\hfill & \mathrm{comp}({(A_0^{†})}_1+{(A_0^{†})}_2)\hfill \\ \hfill =& (\mathrm{comp}{(G_{k-s+1,S})}_1){)}^{2^{3(s-r-1)}-1}\mathrm{comp}({(G_{k-s+1,S})}_1\setminus 0)\hfill \\ \hfill =& \mathrm{comp}({(A_0^{†})}_2).\hfill \end{array}$$

It follows that

$$\begin{array}{cc}\hfill & \mathrm{comp}({(A^{†})}_1+{(A^{†})}_2)\hfill \\ \hfill =& \mathrm{comp}({(A_0^{†})}_1+{(A_0^{†})}_2){(\mathrm{comp}({(A_0)}_1+{(A_0)}_2))}^3{(\mathrm{comp}({(A_1)}_1+{(A_1)}_2))}^4\hfill \\ \hfill =& \mathrm{comp}({(A_0^{†})}_2){(\mathrm{comp}({(A_0)}_2))}^3{(\mathrm{comp}({(A_1)}_2))}^4\hfill \\ \hfill =& \mathrm{comp}({(A^{†})}_2).\hfill \end{array}$$

For Case I, we obtain

$$\begin{array}{cc}\hfill & \mathrm{comp}({(G_{k,h,S}^{\alpha })}_r+{(G_{k,h,S}^{\alpha })}_s)\hfill \\ \hfill =& \mathrm{comp}({(A^{†})}_1+{(A^{†})}_2){(\mathrm{comp}({(A)}_1+{(A)}_2))}^{2^{3(r-1)}-1}\hfill \\ \hfill =& \mathrm{comp}({(A^{†})}_2){(\mathrm{comp}({(A)}_2))}^{2^{3(r-1)}-1}\hfill \\ \hfill =& \mathrm{comp}({(G_{k,h,S}^{\alpha })}_s).\hfill \end{array}$$

The reasoning for the other two instances parallels that of Case I. Consequently, for $1\le r<s\le k$, we have

$$\mathrm{comp}\left({(G_{k,h,S}^{\alpha })}_r+{(G_{k,h,S}^{\alpha })}_s\right)=\mathrm{comp}\left({(G_{k,h,S}^{\alpha })}_s\right).$$

□

As a direct consequence of Proposition 4, the following corollary emerges.

Corollary 1. 

For $1\le r_1<r_2<\cdots <r_m\le k$, we have

$$\mathrm{comp}\left({(G_{k,h,S}^{\alpha })}_{r_1}+{(G_{k,h,S}^{\alpha })}_{r_2}+\cdots +{(G_{k,h,S}^{\alpha })}_{r_m}\right)=\mathrm{comp}\left({(G_{k,h,S}^{\alpha })}_{r_m}\right).$$

By leveraging Proposition 4 and Corollary 1, we can ascertain the complete weight enumerator of ${\mathcal{S}}_{k,h}^{\alpha }$.

Theorem 1. 

The complete weight enumerator of ${\mathcal{S}}_{k,h}^{\alpha }$ is

$$\begin{array}{cc}\hfill & W_{{\mathcal{S}}_{k,h}^{\alpha }}(Y_0,Y_1)\hfill \\ \hfill =& Y_0^{2^{3k}-2^{3h}}+(2^{k-h}-1)Y_0^{2^{3k-1}-2^{3h}}{Y}_1^{2^{3k-1}}+(2^k-2^{k-h})Y_0^{2^{3k-1}-2^{3h-1}}{Y}_1^{2^{3k-1}-2^{3h-1}},\hfill \end{array}$$

where $1\le h\le k-1$.

Proof. 

Given that the torsion code ${\mathcal{S}}_{k,h}^{\alpha }$ constitutes a linear code of length $2^{3k}-2^{3h}$ over ${\mathbb{F}}_2$, and is generated by $G_{k,h,S}^{\alpha }$, it follows that, for any codeword $a\in {\mathcal{S}}_{k,h}^{\alpha }$, one can find $(b_1,b_2,\cdots ,b_k)\in {\mathbb{F}}_2^k$ such that $a=b_1·{(G_{k,h,S}^{\alpha })}_1+b_2·{(G_{k,h,S}^{\alpha })}_2+\cdots +b_k·{(G_{k,h,S}^{\alpha })}_k$. In accordance with Corollary 1, we infer that $comp\left(b_1·{(G_{k,h,S}^{\alpha })}_1+b_2·{(G_{k,h,S}^{\alpha })}_2+\cdots +b_k·{(G_{k,h,S}^{\alpha })}_k\right)=comp\left({(G_{k,h,S}^{\alpha })}_r\right)$, where r indicates the greatest index for which $b_r\ne 0$. Evidently, the quantity of codewords having a composition that matches $comp\left({(G_{k,h,S}^{\alpha })}_r\right)$ is $2^{r-1}$. Therefore, the complete weight enumerator of ${\mathcal{S}}_{k,h}^{\alpha }$ is

$$\begin{array}{cc}\hfill & W_{{\mathcal{S}}_{k,h}^{\alpha }}(Y_0,Y_1)\hfill \\ \hfill =& Y_0^{2^{3k}-2^{3h}}+\left(comp({(G_{k,h,S}^{\alpha })}_1)\right)+2\left(comp({(G_{k,h,S}^{\alpha })}_2)\right)+\cdots +2^{k-1}\left(comp({(G_{k,h,S}^{\alpha })}_k)\right)\hfill \\ \hfill =& Y_0^{2^{3k}-2^{3h}}+(2^{k-h}-1)Y_0^{2^{3k-1}-2^{3h}}{Y}_1^{2^{3k-1}}+(2^k-2^{k-h})Y_0^{2^{3k-1}-2^{3h-1}}{Y}_1^{2^{3k-1}-2^{3h-1}}.\hfill \end{array}$$

□

Denote by $G_{k,S}^{\beta }$, ${\lambda }_{k,S}$, ${\delta }_{k,S}$, ${\sigma }_{k,S}$, ${\varrho }_{k,S}$, ${\tau }_{k,S}$, ${\nu }_{k,S}$ the matrices derived from substituting 1 for $w_1$ in the original matrices $w_1{G}_k^{\beta }$, $w_1{\lambda }_k$, $w_1{\delta }_k$, $u{\sigma }_k$, $w_1{\varrho }_k$, $w_1{\tau }_k$, $w_1{\nu }_k$, respectively. Similarly, let $G_{k,h,S}^{\beta }$ represent the matrix obtained from $w_1{G}_{k,h}^{\beta }$ by substituting 1 for $w_1$, where $G_{k,h,S}^{\beta }$ serves as the generator matrix for the torsion code ${\mathcal{S}}_{k,h}^{\beta }$. Subsequently, we aim to establish the compositions of the r-th rows, denoted as ${\left(G_{k,S}^{\alpha }\right)}_r$ and ${\left(G_{k,h,S}^{\alpha }\right)}_r$, across all indices r from 1 to k.

Lemma 2. 

For any $k\ge 1$, we have

$$\begin{array}{cc}\hfill comp({({\lambda }_{k,S})}_1)& =Y_0^{2^{3k-1}-2^{2k}}{Y}_1^{2^{3k-1}},\hfill \\ \hfill comp({({\delta }_{k,S})}_1)& =Y_0^{2^{3k-1}-2^{2k-1}}{Y}_1^{2^{3k-1}-2^{2k-1}},\hfill \\ \hfill comp({({\sigma }_{k,S})}_1)& =Y_0^{2^{3k-1}-2^{2k-1}}{Y}_1^{2^{3k-1}-2^{2k-1}},\hfill \\ \hfill comp({({\varrho }_{k,S})}_1)& =Y_0^{2^{3k-1}+2^{k-1}-2^{2k}}{Y}_1^{2^{3k-1}+2^{k-1}-2^{2k}},\hfill \\ \hfill comp({({\tau }_{k,S})}_1)& =Y_0^{2^{3k-1}-2^{2k}-2^{2k-1}+2^k}{Y}_1^{2^{3k-1}-2^{2k-1}},\hfill \\ \hfill comp({({\nu }_{k,S})}_1)& =Y_0^{2^{3k-1}-2^{2k}-2^{2k-1}+2^k}{Y}_1^{2^{3k-1}-2^{2k-1}},\hfill \\ \hfill comp({(G_{k,S}^{\beta })}_1)& =Y_0^{2^{3k-1}-2^{2k+1}+2^{k+1}+2^{k-1}-1}{Y}_1^{2^{3k-1}-2^{2k}+2^{k-1}}.\hfill \end{array}$$

Proof. 

For ${\lambda }_k$, $k\ge 1$, we have

$$\begin{array}{c}\hfill {({\lambda }_{k,S})}_1=\left[\begin{array}{cccccccc}0& 1& 1& 1& 1& 0& 0& 0\end{array}\right].\end{array}$$

Therefore,

$$\begin{array}{cc}\hfill comp({({\lambda }_{k,S})}_1)& =Y_0^{4|{\lambda }_{k-1}^{\alpha }|}{Y}_1^{4|G_{k-1}^{\alpha }|}\hfill \\ \hfill & =Y_0^{2^{3k-1}-2^{2k}}{Y}_1^{2^{3k-1}}.\hfill \end{array}$$

For ${\delta }_k$, $k\ge 1$, we have

$$\begin{array}{c}\hfill {({\delta }_{k,S})}_1=\left[\begin{array}{cccccccc}0& 1& 0& 1& 0& 1& 1& 0\end{array}\right].\end{array}$$

Therefore,

$$\begin{array}{cc}\hfill comp({({\delta }_{k,S})}_1)& =Y_0^{2|G_{k-1}^{\alpha }|+2|{\delta }_{k-1}|}{Y}_1^{2|G_{k-1}^{\alpha }|+2|{\delta }_{k-1}|}\hfill \\ \hfill & =Y_0^{2^{3k-1}-2^{2k-1}}{Y}_1^{2^{3k-1}-2^{2k-1}}.\hfill \end{array}$$

For ${\sigma }_k$, $k\ge 1$, we have

$$\begin{array}{c}\hfill {({\sigma }_{k,S})}_1=\left[\begin{array}{cccccccc}0& 1& 1& 0& 0& 1& 0& 1\end{array}\right].\end{array}$$

Therefore,

$$\begin{array}{cc}\hfill comp({({\sigma }_{k,S})}_1)& =Y_0^{2|G_{k-1}^{\alpha }|+2|{\sigma }_{k-1}|}{Y}_1^{2|G_{k-1}^{\alpha }|+2|{\sigma }_{k-1}|}\hfill \\ \hfill & =Y_0^{2^{3k-1}-2^{2k-1}}{Y}_1^{2^{3k-1}-2^{2k-1}}.\hfill \end{array}$$

For ${\varrho }_k$, $k\ge 1$, we have

$$\begin{array}{c}\hfill {({\varrho }_{k,S})}_1=\left[\begin{array}{cccccccc}0& 1& 0& 0& 1& 1& 0& 1\end{array}\right].\end{array}$$

Therefore,

$$\begin{array}{cc}\hfill comp({({\varrho }_{k,S})}_1)& =Y_0^{|{\varrho }_{k-1}|+|G_{k-1}^{\alpha }|+|{\delta }_{k-1}|+|{\sigma }_{k-1}|}{Y}_1^{|G_{k-1}^{\alpha }|+|{\delta }_{k-1}|+|{\sigma }_{k-1}|+|{\varrho }_{k-1}|}\hfill \\ \hfill & =Y_0^{2^{3k-1}+2^{k-1}-2^{2k}}{Y}_1^{2^{3k-1}+2^{k-1}-2^{2k}}.\hfill \end{array}$$

For ${\tau }_k$, $k\ge 1$, we have

$$\begin{array}{c}\hfill {({\tau }_{k,S})}_1=\left[\begin{array}{cccccccc}0& 1& 1& 0& 0& 1& 1& 0\end{array}\right].\end{array}$$

Therefore,

$$\begin{array}{cc}\hfill comp({({\tau }_{k,S})}_1)& =Y_0^{2|{\tau }_{k-1}|+2|{\lambda }_{k-1}|}{Y}_1^{2|G_{k-1}^{\alpha }|+2|{\sigma }_{k-1}|}\hfill \\ \hfill & =Y_0^{2^{3k-1}-2^{2k}-2^{2k-1}+2^k}{Y}_1^{2^{3k-1}-2^{2k-1}}.\hfill \end{array}$$

For ${\nu }_k$, $k\ge 1$, we have

$$\begin{array}{c}\hfill {({\nu }_{k,S})}_1=\left[\begin{array}{cccccccc}0& 1& 1& 0& 0& 1& 1& 0\end{array}\right].\end{array}$$

Therefore,

$$\begin{array}{cc}\hfill comp({({\nu }_{k,S})}_1)& =Y_0^{2|{\nu }_{k-1}|+2|{\lambda }_{k-1}|}{Y}_1^{2|G_{k-1}^{\alpha }|+2|{\delta }_{k-1}|}\hfill \\ \hfill & =Y_0^{2^{3k-1}-2^{2k}-2^{2k-1}+2^k}{Y}_1^{2^{3k-1}-2^{2k-1}}.\hfill \end{array}$$

For $G_k^{\beta }$, $k\ge 1$, we have

$$\begin{array}{c}\hfill {(G_{k,S}^{\beta })}_1=\left[\begin{array}{cccccccc}1& 0& 0& 1& 1& 1& 0& 0\end{array}\right].\end{array}$$

Therefore,

$$\begin{array}{cc}\hfill comp({(G_{k,S}^{\beta })}_1)& =Y_0^{|G_{k-1}^{\beta }|+|{\lambda }_{k-1}|+|{\tau }_{k-1}|+|{\nu }_{k-1}|}{Y}_1^{|G_{k-1}^{\alpha }|+|{\delta }_{k-1}|+|{\sigma }_{k-1}|+|{\varrho }_{k-1}|}\hfill \\ \hfill & =Y_0^{2^{3k-1}-2^{2k+1}+2^{k+1}+2^{k-1}-1}{Y}_1^{2^{3k-1}-2^{2k}+2^{k-1}}.\hfill \end{array}$$

□

From the constructions of $G_k^{\beta }$, ${\lambda }_k$, ${\delta }_k$, ${\sigma }_k$, ${\varrho }_k$, ${\tau }_k$, ${\nu }_k$ and Lemma 2, we can obtain the compositions of ${(G_{k,S}^{\beta })}_r$, ${({\lambda }_{k,S})}_r$, ${({\delta }_{k,S})}_r$, ${({\sigma }_{k,S})}_r$, ${({\varrho }_{k,S})}_r$, ${({\tau }_{k,S})}_r$, ${({\nu }_{k,S})}_r$ for $k\ge 1$, $1\le r\le k$.

Lemma 3. 

For $1\le r\le k$, we have

$$\begin{array}{cc}\hfill comp({({\lambda }_{k,S})}_r)& ={(comp({({\lambda }_{k-r+1,S})}_1))}^{4^{r-1}}{(comp({(G_{k-r+1,S})}_1))}^{8^{r-1}-4^{r-1}},\hfill \\ \hfill comp({({\delta }_{k,S})}_r)& ={(comp({({\delta }_{k-r+1,S})}_1))}^{4^{r-1}}{(comp({(G_{k-r+1,S})}_1))}^{8^{r-1}-4^{r-1}},\hfill \\ \hfill comp({({\sigma }_{k,S})}_r)& ={(comp({({\sigma }_{k-r+1,S})}_1))}^{4^{r-1}}{(comp({(G_{k-r+1,S})}_1))}^{8^{r-1}-4^{r-1}},\hfill \\ \hfill comp({({\varrho }_{k,S})}_r)& ={(comp({({\varrho }_{k-r+1,S})}_1))}^{2^{r-1}}{(comp({({\delta }_{k-r+1,S})}_1))}^{2^{2r-2}-2^{r-1}}\hfill \\ \hfill & {(comp({({\sigma }_{k-r+1,S})}_1))}^{2^{2r-2}-2^{r-1}}{(comp({(G_{k-r+1,S})}_1))}^{2^{r-1}{(2^{r-1}-1)}^2},\hfill \\ \hfill comp({({\tau }_{k,S})}_r)& ={(comp({({\tau }_{k-r+1,S})}_1))}^{2^{r-1}}{(comp({({\lambda }_{k-r+1,S})}_1))}^{2^{2r-2}-2^{r-1}}\hfill \\ \hfill & {(comp({({\sigma }_{k-r+1,S})}_1))}^{2^{2r-2}-2^{r-1}}{(comp({(G_{k-r+1,S})}_1))}^{2^{r-1}{(2^{r-1}-1)}^2},\hfill \\ \hfill comp({({\nu }_{k,S})}_r)& ={(comp({({\nu }_{k-r+1,S})}_1))}^{2^{r-1}}{(comp({({\lambda }_{k-r+1,S})}_1))}^{2^{2r-2}-2^{r-1}}\hfill \\ \hfill & {(comp({({\delta }_{k-r+1,S})}_1))}^{2^{2r-2}-2^{r-1}}{(comp({(G_{k-r+1,S})}_1))}^{2^{r-1}{(2^{r-1}-1)}^2},\hfill \\ \hfill comp({(G_{k,S}^{\beta })}_r)& ={(comp({(G_{k-r+1,S}^{\alpha })}_1))}^{{\displaystyle \sum _{\kappa =1}^{r-1}}{2}^{3(\kappa -1)}+3{\displaystyle \sum _{\kappa =1}^{r-1}}(2^{3\kappa -3}-2^{2\kappa -2})+3{\displaystyle \sum _{\kappa =1}^{r-1}}{2}^{\kappa -1}{(2^{\kappa -1}-1)}^2}\hfill \\ \hfill & comp({(G_{k-r+1,S}^{\beta })}_1){(comp({({\lambda }_{k-r+1,S})}_1))}^{{\displaystyle \sum _{\kappa =1}^{r-1}}{2}^{2(\kappa -1)}+2{\displaystyle \sum _{\kappa =1}^{r-1}}(2^{2\kappa -2}-2^{\kappa -1})}\hfill \\ \hfill & {(comp({({\delta }_{k-r+1,S})}_1))}^{{\displaystyle \sum _{\kappa =1}^{r-1}}{2}^{2(\kappa -1)}+2{\displaystyle \sum _{\kappa =1}^{r-1}}(2^{2\kappa -2}-2^{\kappa -1})}{(comp({({\tau }_{k-r+1,S})}_1))}^{{\displaystyle \sum _{\kappa =1}^{r-1}}{2}^{\kappa -1}}\hfill \\ \hfill & {(comp({({\sigma }_{k-r+1,S})}_1))}^{{\displaystyle \sum _{\kappa =1}^{r-1}}{2}^{2(\kappa -1)}+2{\displaystyle \sum _{\kappa =1}^{r-1}}(2^{2\kappa -2}-2^{\kappa -1})}{(comp({({\nu }_{k-r+1,S})}_1))}^{{\displaystyle \sum _{\kappa =1}^{r-1}}{2}^{\kappa -1}}\hfill \\ \hfill & {(comp({({\varrho }_{k-r+1,S})}_1))}^{{\displaystyle \sum _{\kappa =1}^{r-1}}{2}^{\kappa -1}}.\hfill \end{array}$$

Proof. 

Obviously, these results hold when $r=1$. Subsequently, we will demonstrate that the aforementioned results are applicable for any $2\le r\le k$. Based on the construction of ${\lambda }_k$, the composition of ${({\lambda }_{k,S})}_r$, for $2\le r\le k$, is

$$\begin{array}{cc}\hfill & comp({({\lambda }_{k,S})}_r)\hfill \\ \hfill =& {(comp({({\lambda }_{k-r+1,S})}_1))}^{4^{r-1}}{\displaystyle \prod _{\kappa =1}^{r-1}}{(comp({(G_{k-\kappa ,S})}_{r-\kappa }))}^{4^{\kappa }}\hfill \\ \hfill =& {(comp({({\lambda }_{k-r+1,S})}_1))}^{4^{r-1}}{(comp({(G_{k-r+1,S})}_1))}^{8^{r-1}-4^{r-1}}\hfill \end{array}$$

The verification procedures for the compositions of ${({\delta }_{k,S})}_r$ and ${({\sigma }_{k,S})}_r$ parallel those for ${({\lambda }_{k,S})}_r$.

Additionally, the compositions of ${({\varrho }_{k,S})}_r$, for $1\le r\le k$, is

$$\begin{array}{cc}\hfill & comp({({\varrho }_{k,S})}_r)\hfill \\ \hfill =& {(comp({({\varrho }_{k-r+1,S})}_1))}^{2^{r-1}}{\displaystyle \prod _{\kappa =1}^{r-1}}{(comp({(G_{k-\kappa ,S})}_{r-\kappa }))}^{2^{\kappa }}\hfill \\ \hfill & {\displaystyle \prod _{\kappa =1}^{r-1}}{(comp({({\delta }_{k-\kappa ,S})}_{r-\kappa }))}^{2^{\kappa }}{\displaystyle \prod _{\kappa =1}^{r-1}}{(comp({({\sigma }_{k-\kappa ,S})}_{r-\kappa }))}^{2^{\kappa }}\hfill \\ \hfill =& {(comp({({\varrho }_{k-r+1,S})}_1))}^{2^{r-1}}{(comp({({\delta }_{k-r+1,S})}_1))}^{2^{2r-2}-2^{r-1}}\hfill \\ & {(comp({({\sigma }_{k-r+1,S})}_1))}^{2^{2r-2}-2^{r-1}}{(comp({(G_{k-r+1,S})}_1))}^{2^{r-1}{(2^{r-1}-1)}^2}.\hfill \end{array}$$

The verification procedures for the compositions of ${({\tau }_{k,S})}_r$ and ${({\nu }_{k,S})}_r$ parallel those for ${({\varrho }_{k,S})}_r$.

The compositions of ${(G_{k,S}^{\beta })}_r$, for $1\le r\le k$, is

$$\begin{array}{cc}\hfill & comp({(G_{k,S}^{\beta })}_r)\hfill \\ \hfill =& {\displaystyle \prod _{\kappa =1}^{r-1}}(comp({(G_{k-\kappa ,S}^{\alpha })}_{r-\kappa })){\displaystyle \prod _{\kappa =1}^{r-1}}(comp({({\lambda }_{k-\kappa ,S})}_{r-\kappa })){\displaystyle \prod _{\kappa =1}^{r-1}}(comp({({\delta }_{k-\kappa ,S})}_{r-\kappa }))\hfill \\ \hfill & {\displaystyle \prod _{\kappa =1}^{r-1}}(comp({({\sigma }_{k-\kappa ,S})}_{r-\kappa })){\displaystyle \prod _{\kappa =1}^{r-1}}(comp({({\varrho }_{k-\kappa ,S})}_{r-\kappa })){\displaystyle \prod _{\kappa =1}^{r-1}}(comp({({\tau }_{k-\kappa ,S})}_{r-\kappa }))\hfill \\ \hfill & {\displaystyle \prod _{\kappa =1}^{r-1}}(comp({({\nu }_{k-\kappa ,S})}_{r-\kappa }))comp({(G_{k-r+1,S}^{\beta })}_1)\hfill \\ \hfill =& {(comp({(G_{k-r+1,S}^{\alpha })}_1))}^{{\displaystyle \sum _{\kappa =1}^{r-1}}{2}^{3(\kappa -1)}+3{\displaystyle \sum _{\kappa =1}^{r-1}}(2^{3\kappa -3}-2^{2\kappa -2})+3{\displaystyle \sum _{\kappa =1}^{r-1}}{2}^{\kappa -1}{(2^{\kappa -1}-1)}^2}comp({(G_{k-r+1,S}^{\beta })}_1)\hfill \\ \hfill & {(comp({({\lambda }_{k-r+1,S})}_1))}^{{\displaystyle \sum _{\kappa =1}^{r-1}}{2}^{2(\kappa -1)}+2{\displaystyle \sum _{\kappa =1}^{r-1}}(2^{2\kappa -2}-2^{\kappa -1})}{(comp({({\delta }_{k-r+1,S})}_1))}^{{\displaystyle \sum _{\kappa =1}^{r-1}}{2}^{2(\kappa -1)}+2{\displaystyle \sum _{\kappa =1}^{r-1}}(2^{2\kappa -2}-2^{\kappa -1})}\hfill \\ \hfill & {(comp({({\sigma }_{k-r+1,S})}_1))}^{{\displaystyle \sum _{\kappa =1}^{r-1}}{2}^{2(\kappa -1)}+2{\displaystyle \sum _{\kappa =1}^{r-1}}(2^{2\kappa -2}-2^{\kappa -1})}{(comp({({\varrho }_{k-r+1,S})}_1))}^{{\displaystyle \sum _{\kappa =1}^{r-1}}{2}^{\kappa -1}}\hfill \\ \hfill & {(comp({({\tau }_{k-r+1,S})}_1))}^{{\displaystyle \sum _{\kappa =1}^{r-1}}{2}^{\kappa -1}}{(comp({({\nu }_{k-r+1,S})}_1))}^{{\displaystyle \sum _{\kappa =1}^{r-1}}{2}^{\kappa -1}}.\hfill \end{array}$$

□

Using Lemmas 2 and 3, we can deduce the composition of ${(G_{k,h,S}^{\beta })}_r$. For convenience, let $A_r={\displaystyle \sum _{\kappa =1}^r}{2}^{3(\kappa -1)}+3{\displaystyle \sum _{\kappa =1}^r}(2^{3\kappa -3}-2^{2\kappa -2})+3{\displaystyle \sum _{\kappa =1}^r}{2}^{\kappa -1}{(2^{\kappa -1}-1)}^2$, $B_r={\displaystyle \sum _{\kappa =1}^r}{2}^{2(\kappa -1)}+2{\displaystyle \sum _{\kappa =1}^r}(2^{2\kappa -2}-2^{\kappa -1})$, $C_r={\displaystyle \sum _{\kappa =1}^r}{2}^{\kappa -1}$, $A_0=B_0=C_0=0$.

Proposition 5. 

For any $k\ge 2$, with $1\le h\le k-1$, the composition of ${(G_{k,h,S}^{\beta })}_r$ can be addressed under the three following scenarios.

Case I: When $r=1$,

$$\begin{array}{cc}\hfill & comp({(G_{k,h,S}^{\beta })}_1)\hfill \\ \hfill =& Y_0^{2^{3k-1}-2^{2k+1}+2^{k+1}+2^{k-1}-2^{3h}+3·2^{2h}-3·2^h}{Y}_1^{2^{3k-1}-2^{2k}+2^{k-1}}.\hfill \end{array}$$

Case II: When $2\le r\le k-h$, we have

$$\begin{array}{cc}\hfill & comp({(G_{k,h,S}^{\beta })}_r)\hfill \\ \hfill =& Y_0^{{\phi }_0\left({\left(G_{k,h,S}^{\beta }\right)}_r\right)}{Y}_1^{{\phi }_1\left({\left(G_{k,h,S}^{\beta }\right)}_r\right)},\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill & {\phi }_0\left({\left(G_{k,h,S}^{\beta }\right)}_r\right)\hfill \\ \hfill =& 2^{3k-3r+2}{A}_{r-1}+(3·2^{3k-3r+2}-2^{2k-2r+3})B_{r-1}+(3·2^{3k-3r+2}+5·2^{k-r}-2^{2k-2r+4})C_{r-1}\hfill \\ \hfill & +2^{3k-3r+2}-2^{2k-2r+3}+5·2^{k-r}-2^{3h}+3·2^{2h}-3·2^h,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\phi }_1\left({\left(G_{k,h,S}^{\beta }\right)}_r\right)\hfill \\ \hfill =& 2^{3k-3r+2}{A}_{r-1}+(3·2^{3k-3r+2}-2^{2k-2r+2})B_{r-1}+(3·2^{3k-3r+2}-2^{2k-2r+3}+2^{k-r})C_{r-1}\hfill \\ \hfill & +2^{3k-3r+2}-2^{2k-2r+2}+2^{k-r}.\hfill \end{array}$$

Case III: When $k-h+1\le r\le k$, we have

$$\begin{array}{cc}\hfill & comp({(G_{k,h,S}^{\beta })}_r)\hfill \\ \hfill =& Y_0^{{\phi }_0\left({\left(G_{k,h,S}^{\beta }\right)}_r\right)}{Y}_1^{{\phi }_1\left({\left(G_{k,h,S}^{\beta }\right)}_r\right)},\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill & {\phi }_0\left({\left(G_{k,h,S}^{\beta }\right)}_r\right)\hfill \\ \hfill =& 2^{3k-3r+2}(A_{r-1}-A_{r-k+h-1})+(3·2^{3k-3r+2}-2^{2k-2r+3})(B_{r-1}-B_{r-k+h-1})\hfill \\ \hfill & +(3·2^{3k-3r+2}+5·2^{k-r}-2^{2k-2r+4})(C_{r-1}-C_{r-k+h-1}),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\phi }_1\left({\left(G_{k,h,S}^{\beta }\right)}_r\right)\hfill \\ \hfill =& 2^{3k-3r+2}(A_{r-1}-A_{r-k+h-1})+(3·2^{3k-3r+2}-2^{2k-2r+2})(B_{r-1}-B_{r-k+h-1})\hfill \\ \hfill & +(3·2^{3k-3r+2}-2^{2k-2r+3}+2^{k-r})(C_{r-1}-C_{r-k+h-1}).\hfill \end{array}$$

Lemma 4. 

For $k\ge 2$, $1\le r<s\le k$, we have

$$\begin{array}{cc}\hfill & \mathrm{comp}\left({({\lambda }_{k,S})}_r+{({\lambda }_{k,S})}_s\right)=\mathrm{comp}\left({({\lambda }_{k,S})}_s\right),\hfill \\ \hfill & \mathrm{comp}\left({({\delta }_{k,S})}_r+{({\delta }_{k,S})}_s\right)=\mathrm{comp}\left({({\delta }_{k,S})}_s\right),\hfill \\ \hfill & \mathrm{comp}\left({({\sigma }_{k,S})}_r+{({\sigma }_{k,S})}_s\right)=\mathrm{comp}\left({({\sigma }_{k,S})}_s\right),\hfill \\ \hfill & \mathrm{comp}\left({({\varrho }_{k,S})}_r+{({\varrho }_{k,S})}_s\right)=\mathrm{comp}\left({({\varrho }_{k,S})}_s\right),\hfill \\ \hfill & \mathrm{comp}\left({({\tau }_{k,S})}_r+{({\tau }_{k,S})}_s\right)=\mathrm{comp}\left({({\tau }_{k,S})}_s\right),\hfill \\ \hfill & \mathrm{comp}\left({({\nu }_{k,S})}_r+{({\nu }_{k,S})}_s\right)=\mathrm{comp}\left({({\nu }_{k,S})}_s\right),\hfill \\ \hfill & \mathrm{comp}\left({(G_{k,S}^{\beta })}_r+{(G_{k,S}^{\beta })}_s\right)=\mathrm{comp}\left({(G_{k,S}^{\beta })}_s\right).\hfill \end{array}$$

Proof. 

Clearly, these results are valid for $k=2,r=1,s=2$. Suppose that

$$\mathrm{comp}\left({({\lambda }_{k-1,S})}_r+{({\lambda }_{k-1,S})}_s\right)=\mathrm{comp}\left({({\lambda }_{k-1,S})}_s\right)$$

is valid for any $1\le r<s\le k-1$. When $r=1,2\le s\le k$, it is easy to verify that

$$\mathrm{comp}\left({({\lambda }_{k,S})}_1+{({\lambda }_{k,S})}_s\right)=\mathrm{comp}\left({({\lambda }_{k,S})}_s\right).$$

When $2\le r<s\le k$, we have that

$$\begin{array}{cc}\hfill & comp\left({({\lambda }_{k,S})}_r+{({\lambda }_{k,S})}_s\right)\hfill \\ \hfill =& {(comp\left({({\lambda }_{k-1,S})}_{r-1}+{({\lambda }_{k-1,S})}_{s-1}\right))}^4{(comp\left({(G_{k-1,S})}_{r-1}+{(G_{k-1,S})}_{s-1}\right))}^4\hfill \\ \hfill =& {(comp\left({({\lambda }_{k-1,S})}_{s-1}\right))}^4{(comp\left({(G_{k-1,S})}_{s-1}\right))}^4\hfill \\ \hfill =& comp\left({({\lambda }_{k,S})}_s\right).\hfill \end{array}$$

Therefore, for any $1\le r<s\le k$, we have

$$comp\left({({\lambda }_{k,S})}_r+{({\lambda }_{k,S})}_s\right)=comp\left({({\lambda }_{k,S})}_s\right).$$

The verification procedures for the compositions of ${({\delta }_{k,S})}_r+{({\delta }_{k,S})}_s$ and ${({\sigma }_{k,S})}_r+{({\sigma }_{k,S})}_s$ parallel those for ${({\lambda }_{k,S})}_r+{({\lambda }_{k,S})}_s$.

Suppose that

$$\mathrm{comp}\left({({\varrho }_{k-1,S})}_r+{({\varrho }_{k-1,S})}_s\right)=\mathrm{comp}\left({({\varrho }_{k-1,S})}_s\right)$$

is valid for any $1\le r<s\le k-1$. When $r=1,2\le s\le k$, it is easy to verify that

$$\mathrm{comp}\left({({\varrho }_{k,S})}_1+{({\varrho }_{k,S})}_s\right)=\mathrm{comp}\left({({\varrho }_{k,S})}_s\right).$$

When $2\le r<s\le k$, we have that

$$\begin{array}{cc}\hfill & comp\left({({\varrho }_{k,S})}_r+{({\varrho }_{k,S})}_s\right)\hfill \\ \hfill =& {(comp\left({({\varrho }_{k-1,S})}_{r-1}+{({\varrho }_{k-1,S})}_{s-1}\right))}^2{(comp\left({(G_{k-1,S})}_{r-1}+{(G_{k-1,S})}_{s-1}\right))}^2\hfill \\ \hfill & {(comp\left({({\delta }_{k-1,S})}_{r-1}+{({\delta }_{k-1,S})}_{s-1}\right))}^2{(comp\left({({\sigma }_{k-1,S})}_{r-1}+{({\sigma }_{k-1,S})}_{s-1}\right))}^2\hfill \\ \hfill =& {(comp\left({({\varrho }_{k-1,S})}_{s-1}\right))}^2{(comp\left({(G_{k-1,S})}_{s-1}\right))}^2\hfill \\ \hfill & {(comp\left({({\delta }_{k-1,S})}_{s-1}\right))}^2{(comp\left({({\sigma }_{k-1,S})}_{s-1}\right))}^2\hfill \\ \hfill =& comp\left({({\varrho }_{k,S})}_s\right).\hfill \end{array}$$

Therefore, for any $1\le r<s\le k$, we have

$$comp\left({({\varrho }_{k,S})}_r+{({\lambda }_{k,S})}_s\right)=comp\left({({\varrho }_{k,S})}_s\right).$$

The verification procedures for the compositions of ${({\tau }_{k,S})}_r+{({\tau }_{k,S})}_s$ and ${({\nu }_{k,S})}_r+{({\nu }_{k,S})}_s$ parallel those for ${({\varrho }_{k,S})}_r+{({\varrho }_{k,S})}_s$.

Suppose that

$$\mathrm{comp}\left({(G_{k-1,S}^{\beta })}_r+{(G_{k-1,S}^{\beta })}_s\right)=\mathrm{comp}\left({(G_{k-1,S}^{\beta })}_s\right)$$

is valid for any $1\le r<s\le k-1$. When $r=1,2\le s\le k$, it is easy to verify that

$$\mathrm{comp}\left({(G_{k,S}^{\beta })}_1+{(G_{k,S}^{\beta })}_s\right)=\mathrm{comp}\left({(G_{k,S}^{\beta })}_s\right).$$

When $2\le r<s\le k$, we have that

$$\begin{array}{cc}\hfill & comp\left({(G_{k,S}^{\beta })}_r+{(G_{k,S}^{\beta })}_s\right)\hfill \\ \hfill =& comp\left({(G_{k-1,S}^{\alpha })}_{r-1}+{(G_{k-1,S}^{\alpha })}_{s-1}\right)comp\left({({\lambda }_{k-1,S})}_{r-1}+{({\lambda }_{k-1,S})}_{s-1}\right)\hfill \\ \hfill & comp\left({({\delta }_{k-1,S})}_{r-1}+{({\delta }_{k-1,S})}_{s-1}\right)comp\left({({\sigma }_{k-1,S})}_{r-1}+{({\sigma }_{k-1,S})}_{s-1}\right)\hfill \\ \hfill & comp\left({({\varrho }_{k-1,S})}_{r-1}+{({\varrho }_{k-1,S})}_{s-1}\right)comp\left({({\tau }_{k-1,S})}_{r-1}+{({\tau }_{k-1,S})}_{s-1}\right)\hfill \\ \hfill & comp\left({({\nu }_{k-1,S})}_{r-1}+{({\nu }_{k-1,S})}_{s-1}\right)comp\left({(G_{k-1,S}^{\beta })}_{r-1}+{(G_{k-1,S}^{\beta })}_{s-1}\right)\hfill \\ \hfill =& comp\left({(G_{k-1,S}^{\alpha })}_{s-1}\right)comp\left({({\lambda }_{k-1,S})}_{s-1}\right)comp\left({({\delta }_{k-1,S})}_{s-1}\right)\hfill \\ \hfill & comp\left({({\sigma }_{k-1,S})}_{s-1}\right)comp\left({({\varrho }_{k-1,S})}_{s-1}\right)comp\left({({\tau }_{k-1,S})}_{s-1}\right)\hfill \\ \hfill & comp\left({({\nu }_{k-1,S})}_{s-1}\right)comp\left({(G_{k-1,S}^{\beta })}_{s-1}\right)\hfill \\ \hfill =& comp\left({(G_{k,S}^{\beta })}_s\right).\hfill \end{array}$$

Therefore, for any $1\le r<s\le k$, we have

$$comp\left({(G_{k,S}^{\beta })}_r+{(G_{k,S}^{\beta })}_s\right)=comp\left({(G_{k,S}^{\beta })}_s\right).$$

□

According to Lemma 4, the following results can be obtained.

Proposition 6. 

For any $k\ge 2$, $1\le r<s\le k$, we have

$$\mathrm{comp}\left({(G_{k,h,S}^{\beta })}_r+{(G_{k,h,S}^{\beta })}_s\right)=\mathrm{comp}\left({(G_{k,h,S}^{\beta })}_s\right).$$

Proof. 

Case I: When $1\le r<s\le k-t$, we have

$$\begin{array}{cc}\hfill & comp\left({(G_{k,h,S}^{\beta })}_r+{(G_{k,h,S}^{\beta })}_s\right)\hfill \\ \hfill =& comp\left({(G_{k,S}^{\beta })}_r+{(G_{k,S}^{\beta })}_s\right)Y_0^{-|G_h^{\beta }|}\hfill \\ \hfill =& comp\left({(G_{k,h,S}^{\beta })}_s\right).\hfill \end{array}$$

Case II: When $1\le r\le k-h$, $k-h+1\le s\le k$, we have

$$\begin{array}{cc}\hfill & comp\left({(G_{k,h,S}^{\beta })}_r+{(G_{k,h,S}^{\beta })}_s\right)\hfill \\ \hfill =& comp\left({(G_{k,S}^{\beta })}_r+{(G_{k,S}^{\beta })}_s\right){\left(comp\left({(G_{h,S}^{\beta })}_{s-k+h}\right)\right)}^{-1}\hfill \\ \hfill =& comp\left({(G_{k,h,S}^{\beta })}_s\right).\hfill \end{array}$$

Case III: When $k-h+1\le r<s\le k$, we have

$$\begin{array}{cc}\hfill & comp\left({(G_{k,h,S}^{\beta })}_r+{(G_{k,h,S}^{\beta })}_s\right)\hfill \\ \hfill =& comp\left({(G_{k,S}^{\beta })}_r+{(G_{k,S}^{\beta })}_s\right){\left(comp\left({(G_{h,S}^{\beta })}_{r-k+h}+{(G_{h,S}^{\beta })}_{s-k+h}\right)\right)}^{-1}\hfill \\ \hfill =& comp\left({(G_{k,h,S}^{\beta })}_s\right).\hfill \end{array}$$

□

Directly from Proposition 6, the following corollary can be obtained.

Corollary 2. 

For $1\le r_1<r_2<\cdots <r_m\le k$, we have

$$\mathrm{comp}\left({(G_{k,h,S}^{\beta })}_{r_1}+{(G_{k,h,S}^{\beta })}_{r_2}+\cdots +{(G_{k,h,S}^{\beta })}_{r_m}\right)=\mathrm{comp}\left({(G_{k,h,S}^{\beta })}_{r_m}\right).$$

Theorem 2. 

The complete weight enumerator of ${\mathcal{S}}_{k,h}^{\beta }$ is

$$W_{{\mathcal{S}}_{k,h}^{\beta }}(Y_0,Y_1)=Y_0^{|G_{k,h}^{\beta }|}+{\displaystyle \sum _{r=1}^k}{2}^{r-1}comp({(G_{k,h,S}^{\beta })}_r).$$

Proof. 

Given that ${\mathcal{S}}_{k,h}^{\beta }$ constitutes a linear code over ${\mathbb{F}}_2$, and is generated by $G_{k,h,S}^{\beta }$, it follows that, for any codeword $a\in {\mathcal{C}}_{k,h,S}^{\beta }$, one can find $(b_1,b_2,\cdots ,b_k)\in {\mathbb{F}}_2^k$ such that $a=b_1·{(G_{k,h,S}^{\beta })}_1+b_2·{(G_{k,h,S}^{\beta })}_2+\cdots +b_k·{(G_{k,h,S}^{\beta })}_k$. Let r represent the highest index for which $b_r\ne 0$. In accordance with Corollary 2, we infer that

$$comp\left(b_1·{(G_{k,h,S}^{\beta })}_1+b_2·{(G_{k,h,S}^{\beta })}_2+\cdots +b_k·{(G_{k,h,S}^{\beta })}_k\right)=comp\left({(G_{k,h,S}^{\beta })}_r\right).$$

Evidently, the quantity of codewords with a composition that matches $comp\left({(G_{k,h,S}^{\alpha })}_r\right)$ is $2^{r-1}$. Therefore, the complete weight enumerator of ${\mathcal{S}}_{k,h}^{\beta }$ is

$$\begin{array}{c}\hfill W_{{\mathcal{S}}_{k,h}^{\beta }}(Y_0,Y_1)=Y_0^{|G_{k,h}^{\beta }|}+{\displaystyle \sum _{r=1}^k}{2}^{r-1}comp({(G_{k,h,S}^{\beta })}_r).\end{array}$$

□

Example 1. 

Let $k=2$, $h=1$. Using Proposition 5, we have

$$\begin{array}{cc}\hfill & comp({(G_{2,1,S}^{\beta })}_1)\hfill \\ \hfill =& Y_0^{2^{3\times 2-1}-2^{2\times 2+1}+2^{2+1}+2^{2-1}-2^{3\times 1}+3·2^{2\times 1}-3·2^1}{Y}_1^{2^{3\times 2-1}-2^{2\times 2}+2^{2-1}}\hfill \\ \hfill =& Y_0^8{Y}_1^{18}.\hfill \end{array}$$

$$\begin{array}{cc}\hfill & comp({(G_{2,1,S}^{\beta })}_2)\hfill \\ \hfill =& Y_0^{{\phi }_0\left({\left(G_{2,1,S}^{\beta }\right)}_r\right)}{Y}_1^{{\phi }_1\left({\left(G_{2,1,S}^{\beta }\right)}_r\right)}\hfill \\ \hfill =& Y_0^{4·(A_1-A_0)+4·(B_1-B_0)+(C_1-C_0)}{Y}_1^{4·(A_1-A_0)+8·(B_1-B_0)+5·(C_1-C_0)}\hfill \\ \hfill =& Y_0^{4·(1-0)+4·(1-0)+(1-0)}{Y}_1^{4·(1-0)+8·(1-0)+5·(1-0)}\hfill \\ \hfill =& Y_0^9{Y}_1^{17}.\hfill \end{array}$$

From Theorem 2, we derive the complete weight enumerator of ${\mathcal{S}}_{2,1}^{\beta }$ as below

$$\begin{array}{cc}\hfill W_{{\mathcal{S}}_{2,1}^{\beta }}(Y_0,Y_1)=& Y_0^{|G_{2,1}^{\beta }|}+{\displaystyle \sum _{r=1}^2}{2}^{r-1}comp({(G_{2,1,S}^{\beta })}_r\hfill \\ \hfill =& Y_0^{26}+2Y_0^9{Y}_1^{17}+Y_0^8{Y}_1^{18}.\hfill \end{array}$$

Example 2. 

Let $k=3$, $h=2$. Using Theorem 2, we derive the complete weight enumerator of ${\mathcal{S}}_{3,2}^{\beta }$ as below

$$\begin{array}{c}\hfill W_{{\mathcal{S}}_{3,2}^{\beta }}(Y_0,Y_1)=Y_0^{316}+6Y_0^{138}{Y}_1^{178}+Y_0^{120}{Y}_1^{196}.\end{array}$$

4. Applications

This section delves into the applications of the complete weight enumerators of ${\mathcal{S}}_{k,h}^{\alpha }$ and ${\mathcal{S}}_{k,h}^{\beta }$ in the secret sharing schemes and systematic authentication codes.

Massey in [27] outlined the situation of constructing secret sharing schemes employing linear codes over finite field ${\mathbb{F}}_q$. Consider $\mathcal{C}$ as an ${[m,k]}_q$ linear code. Assume that $G=[{\mathit{f}}_0,{\mathit{f}}_1,\dots ,{\mathit{f}}_{m-1}]$ is the generator matrix of $\mathcal{C}$. In the secret sharing scheme relying on $\mathcal{C}$, the secret $e\in {\mathbb{F}}_q$. There are $m-1$ participants, namely $P_l$, $l=1,2,\dots ,m-1$, and a dealer involved.

For the purpose of calculating the shares related to e, the dealer randomly picks a vector $\mathit{s}\in {\mathbb{F}}_q^k$ such that $e=\mathit{s}{\mathit{f}}_0$. After that, the dealer takes $\mathit{s}$ as an information vector and calculates the relevant codeword $\mathit{g}=(g_0,g_1,\dots ,g_{m-1})=\mathit{s}G$ and presents $g_l$ to participant $P_l$ as share for $l=1,2,\dots ,m-1$. Notice that $g_0=\mathit{s}{\mathit{f}}_0=e$. Plainly, a set of shares $\{g_{l_1},g_{l_2},\dots ,g_{l_k}\}$, $1\le l_1<\cdots <l_k\le m-1$ and $1\le k\le m-1$, decides the secret if and only if ${\mathit{f}}_0$ can be linearly represented by ${\mathit{f}}_{l_1},{\mathit{f}}_{l_2},\dots ,{\mathit{f}}_{l_k}$.

Lemma 5 

([27]). Assume that G is the generator matrix of an ${[m,k]}_q$ linear code $\mathcal{C}$. In the secret sharing scheme relying on $\mathcal{C}$, a set of shares $\{g_{l_1},g_{l_2},\dots ,g_{l_k}\}$, $1\le l_1<\cdots <l_k\le m-1$ and $1\le k\le m-1$, decides the secret if and only if there exists a codeword $(1,0,\dots ,0,d_{l_1},0,\dots ,0,d_{l_k},\dots ,0)\in {\mathcal{C}}^{\perp }$, where $d_{l_t}\ne 0$ for no less than one t.

Let D represent a set of participants. A subset $D^{*}\subseteq D$ can restore the secret by putting together their shares, any $D^{\prime }\supseteq D^{*}$ must also be able to restore the secret. $D^{*}$ is regarded as a minimal access set if $D^{*}$ can restore the secret while none of the subsets of $D^{*}$ can. In view of these facts, our focus is solely on the set of all minimal access sets. To identify this set, we should bring in the concept of minimal codewords.

Given a vector $\mathit{d}=(d_0,d_1,\dots ,d_{m-1})$, the support of $\mathit{d}$ is specified as the set $\{0\le l\le m-1:d_l\ne 0\}$. A codeword ${\mathit{d}}_1$ covers codeword ${\mathit{d}}_2$ if the support of ${\mathit{d}}_1$ includes that of ${\mathit{d}}_2$. A non-zero codeword $\mathit{d}$ is considered minimal if it covers only its scalar multiples and none other. From Lemma 5, the set of minimal access sets and the set of minimal codewords of the dual code ${\mathcal{C}}^{\perp }$ with the first component being 1 are in one-to-one correspondence. The covering problem of a linear code $\mathcal{C}$ aims to find all its minimal codewords, which is generally a difficult task. Despite this, it can be worked out for some special linear codes. Ding et al. in [28] outlines the characteristics of the minimal access sets of the secret sharing scheme relying on ${\mathcal{C}}^{\perp }$.

Lemma 6 

([28]). Suppose $\mathcal{C}$ is an ${[m,k]}_2$ linear code and $G=[{\mathit{f}}_0,{\mathit{f}}_1,\dots ,{\mathit{f}}_{m-1}]$ is its generator matrix. For any $0\ne c\in \mathcal{C}$, if c is minimal, then the secret sharing scheme utilizing ${\mathcal{C}}^{\perp }$ will have exactly $2^{k-1}$ minimal access sets. Additionally, the following holds true:

1. 

Provided that ${\mathit{f}}_l$ is a multiple of ${\mathit{f}}_0$, participant $P_l$ must be included in every minimal access set, $1\le l\le m-1$.

2. 

Provided that ${\mathit{f}}_l$ is not a multiple of ${\mathit{f}}_0$, participant $P_l$ must be included in $2^{k-2}$ among the $2^{k-1}$ minimal access sets, $1\le l\le m-1$.

Moving forward, we will delve into the development of secret sharing schemes depending on the dual codes of torsion codes. Firstly, we must establish that every codeword in the torsion codes ${\mathcal{S}}_{k,h}^{\alpha }$ and ${\mathcal{S}}_{k,h}^{\beta }$ is minimal.

Lemma 7 

([29]). Suppose $\mathcal{C}$ is an ${[m,k]}_2$ linear code. Denote $w_{min}$ and $w_{max}$ as the minimum and maximum non-zero weights, respectively. If ${\displaystyle \frac{w_{min}}{w_{max}}}>{\displaystyle \frac{1}{2}}$, then any $c\in \mathcal{C}$ is minimal.

Consider $G_{k,h,S}^{\alpha }$ and $G_{k,h,S}^{\beta }$ as the generator matrices for ${\mathcal{S}}_{k,h}^{\alpha }$ and ${\mathcal{S}}_{k,h}^{\beta }$, respectively. The following results are obtained.

Theorem 3. 

For the secret sharing scheme that is generated from ${{\mathcal{S}}_{k,h}^{\alpha }}^{\perp }$, there exist $2^{3k}-2^{3h}-1$ participants and $2^{k-1}$ minimal access sets. In case the l-th column of $G_{k,h,S}^{\alpha }$ is a multiple of the 0-th column, participant $P_l$ is contained in every minimal access set. In the contrary situation, participant $P_l$ is included in $2^{k-2}$ among $2^{k-1}$ minimal access sets.

Proof. 

We use $w_{min}$ to represent the minimum weight and $w_{max}$ to represent the maximum weight. According to Theorem 1, $w_{min}=2^{3k-1}-2^{3h-1}$ and $w_{max}=2^{3k-1}$. Then, we have

$$\begin{array}{c}\hfill {\displaystyle \frac{w_{min}}{w_{max}}}={\displaystyle \frac{2^{3k-1}-2^{3h-1}}{2^{3k-1}}}=1-{\displaystyle \frac{1}{2^{3k-3h}}}>1-{\displaystyle \frac{1}{2}}={\displaystyle \frac{1}{2}}.\end{array}$$

where $2\le h\le k-1$. From Lemma 7, we deduce that every codeword in ${\mathcal{S}}_{k,h}^{\alpha }$ is minimal. This conclusion is further supported by Lemma 6. □

Theorem 4. 

For the secret sharing scheme that is generated from ${{\mathcal{S}}_{k,h}^{\beta }}^{\perp }$, there exist ${(2^k-1)}^3-{(2^h-1)}^3-1$ participants and $2^{k-1}$ minimal access sets. In case the l-th column of $G_{k,h,S}^{\beta }$ is a multiple of the 0-th column, participant $P_l$ is contained in every minimal access set. In the contrary situation, participant $P_l$ is included in $2^{k-2}$ among $2^{k-1}$ minimal access sets.

Proof. 

We use $w_{min}$ to represent the minimum weight and $w_{max}$ to represent the maximum weight. According to Theorem 2, we obtain $w_{min}=2^{3k-1}-2^{2k}+2^{k-1}-2^{3h-1}+2^{2h}-2^{h-1}$ and $w_{max}=2^{3k-1}-2^{2k}+2^{k-1}$. Then

$$\begin{array}{c}\hfill {\displaystyle \frac{w_{min}}{w_{max}}}={\displaystyle \frac{2^{3k-1}-2^{2k}+2^{k-1}-2^{3h-1}+2^{2h}-2^{h-1}}{2^{3k-1}-2^{2k}+2^{k-1}}}=1-{\displaystyle \frac{1}{2^{k-h}{(\frac{2^k-1}{2^h-1})}^2}}>1-{\displaystyle \frac{1}{2}}={\displaystyle \frac{1}{2}}.\end{array}$$

where $2\le h\le k-1$. From Lemma 7, we deduce that that every codeword in ${\mathcal{S}}_{k,h}^{\beta }$ is minimal. This conclusion is further supported by Lemma 6. □

Remark 1. 

As we know, constructing secret sharing schemes based on minimal codes presents the following two issues: security limitations and lack of scalability and flexibility. Although minimal codes have limitations in constructing secret sharing schemes, their theoretical significance, simplicity, and potential for optimization continue to make them valuable for research.

Example 3. 

We consider the torsion code ${\mathcal{S}}_{3,2}^{\beta }$.

$$G_2^{\beta }=\left[\begin{array}{cccccccc}1& 0& 1+w_1& 1+w_2& w_1+w_2& w_1& w_2& 1+w_1+w_2\\ G_1^{\alpha }& 1& {\lambda }_1& {\delta }_1& {\sigma }_1& {\varrho }_1& {\tau }_1& {\nu }_1\end{array}\right],$$

where

$$\begin{array}{cc}\hfill & G_1^{\alpha }=[01w_1{w}_{2}1+w_{1}1+w_2{w}_1+w_{2}1+w_1+w_2],\hfill \\ \hfill & {\lambda }_1=\left[\begin{array}{cccc}1& 1+w_2& w_1+w_2& w_1\end{array}\right],\hfill \\ \hfill & {\delta }_1=\left[\begin{array}{cccc}1& 1+w_1& w_1+w_2& w_2\end{array}\right],\hfill \\ \hfill & {\sigma }_1=\left[\begin{array}{cccc}1& 1+w_2& 1+w_1& 1+w_1+w_2\end{array}\right],\hfill \\ \hfill & {\varrho }_1=\left[\begin{array}{cc}1& 1+w_1\end{array}\right],\hfill \\ \hfill & {\tau }_1=\left[\begin{array}{cc}1& 1+w_2\end{array}\right],\hfill \\ \hfill & {\nu }_1=\left[\begin{array}{cc}1& w_1+w_2\end{array}\right].\hfill \end{array}$$

$$G_3^{\beta }=\left[\begin{array}{cccccccc}1& 0& 1+w_1& 1+w_2& w_1+w_2& w_1& w_2& 1+w_1+w_2\\ G_2^{\alpha }& G_2^{\beta }& {\lambda }_2& {\delta }_2& {\sigma }_2& {\varrho }_2& {\tau }_2& {\nu }_2\end{array}\right],$$

where

$$\begin{array}{cc}\hfill & G_2^{\alpha }=\left[\begin{array}{cccccccc}0& 1& w_1& w_2& 1+w_1& 1+w_2& w_1+w_2& 1+w_1+w_2\\ G_1^{\alpha }& G_1^{\alpha }& G_1^{\alpha }& G_1^{\alpha }& G_1^{\alpha }& G_1^{\alpha }& G_1^{\alpha }& G_1^{\alpha }\end{array}\right],\hfill \\ \hfill & {\lambda }_2=\left[\begin{array}{cccccccc}0& 1& 1+w_2& w_1+w_2& w_1& w_2& 1+w_1& 1+w_1+w_2\\ {\lambda }_1& G_1^{\alpha }& G_1^{\alpha }& G_1^{\alpha }& G_1^{\alpha }& {\lambda }_1& {\lambda }_1& {\lambda }_1\end{array}\right],\hfill \\ \hfill & {\delta }_2=\left[\begin{array}{cccccccc}0& 1& 1+w_1& w_1+w_2& w_2& w_1& 1+w_2& 1+w_2+w_1\\ {\delta }_1& G_1^{\alpha }& G_1^{\alpha }& G_1^{\alpha }& G_1^{\alpha }& {\delta }_1& {\delta }_1& {\delta }_1\end{array}\right],\hfill \\ \hfill & {\sigma }_2=\left[\begin{array}{cccccccc}0& 1& 1+w_2& 1+w_1& 1+w_1+w_2& w_1& w_2& w_1+w_2\\ {\sigma }_1& G_1^{\alpha }& G_1^{\alpha }& G_1^{\alpha }& G_1^{\alpha }& {\sigma }_1& {\sigma }_1& {\sigma }_1\end{array}\right],\hfill \\ \hfill & {\varrho }_2=\left[\begin{array}{cccccccc}0& 1& 1+w_1& w_2& 1+w_2& w_1& 1+w_1& w_1\\ {\varrho }_1& G_1^{\alpha }& G_1^{\alpha }& {\delta }_1& {\delta }_1& {\sigma }_1& {\sigma }_1& {\varrho }_1\end{array}\right],\hfill \\ \hfill & {\tau }_2=\left[\begin{array}{cccccccc}0& 1& 1+w_2& w_2& 1+w_1& w_1& 1+w_2& w_2\\ {\tau }_1& G_1^{\alpha }& G_1^{\alpha }& {\lambda }_1& {\lambda }_1& {\sigma }_1& {\sigma }_1& {\tau }_1\end{array}\right],\hfill \\ \hfill & {\nu }_2=\left[\begin{array}{cccccccc}0& 1& w_1+w_2& 1+w_1& w_2& 1+w_2& w_1& 1+w_1+w_2\\ {\nu }_1& G_1^{\alpha }& G_1^{\alpha }& {\lambda }_1& {\lambda }_1& {\delta }_1& {\delta }_1& {\nu }_1\end{array}\right].\hfill \end{array}$$

Let $G_{k,S}^{\alpha }$ be the matrix replacing $w_1$ by 1 in $w_1{G}_k^{\alpha }$ and $G_{k,S}^{\beta }$ be the matrix replacing $w_1$ by 1 in $w_1{G}_k^{\beta }$. Moreover, let ${\lambda }_{k,S}$, ${\delta }_{k,S}$, ${\sigma }_{k,S}$, ${\varrho }_{k,S}$, ${\tau }_{k,S}$, and ${\nu }_{k,S}$ be the matrices replacing $w_1$ by 1 in $w_1{\lambda }_k$, $w_1{\delta }_k$, $w_1{\sigma }_k$, $w_1{\varrho }_k$, $w_1{\tau }_k$ and $w_1{\nu }_k$, respectively. Then, we have

$$\begin{array}{cc}\hfill & G_{1,S}^{\alpha }=\left[01100110\right],\hfill \\ \hfill & G_{2,S}^{\alpha }=\left[\begin{array}{cccccccc}0& 1& 1& 0& 0& 1& 1& 0\\ G_{1,S}^{\alpha }& G_{1,S}^{\alpha }& G_{1,S}^{\alpha }& G_{1,S}^{\alpha }& G_{1,S}^{\alpha }& G_{1,S}^{\alpha }& G_{1,S}^{\alpha }& G_{1,S}^{\alpha }\end{array}\right],\hfill \\ \hfill & G_{2,S}^{\beta }=\left[\begin{array}{cccccccc}1& 0& 0& 1& 1& 1& 0& 0\\ G_{1,S}^{\alpha }& 1& {\lambda }_{1,S}& {\delta }_{1,S}& {\sigma }_{1,S}& {\varrho }_{1,S}& {\tau }_{1,S}& {\nu }_{1,S}\end{array}\right],\hfill \\ \hfill & {\lambda }_{1,S}=\left[1111\right],{\delta }_{1,S}=\left[1010\right],{\sigma }_{1,S}=\left[1100\right],\hfill \\ \hfill & {\varrho }_{1,S}=\left[10\right],{\tau }_{1,S}=\left[11\right],{\nu }_{1,S}=\left[11\right],\hfill \\ \hfill & {\lambda }_{2,S}=\left[\begin{array}{cccccccc}0& 1& 1& 1& 1& 0& 0& 0\\ {\lambda }_{1,S}& G_{1,S}^{\alpha }& G_{1,S}^{\alpha }& G_{1,S}^{\alpha }& G_{1,S}^{\alpha }& {\lambda }_{1,S}& {\lambda }_{1,S}& {\lambda }_{1,S}\end{array}\right],\hfill \\ \hfill & {\delta }_{2,S}=\left[\begin{array}{cccccccc}0& 1& 0& 1& 0& 1& 1& 0\\ {\delta }_{1,S}& G_{1,S}^{\alpha }& G_{1,S}^{\alpha }& G_{1,S}^{\alpha }& G_{1,S}^{\alpha }& {\delta }_{1,S}& {\delta }_{1,S}& {\delta }_{1,S}\end{array}\right],\hfill \\ \hfill & {\sigma }_{2,S}=\left[\begin{array}{cccccccc}0& 1& 1& 0& 0& 1& 0& 1\\ {\sigma }_{1,S}& G_{1,S}^{\alpha }& G_{1,S}^{\alpha }& G_{1,S}^{\alpha }& G_{1,S}^{\alpha }& {\sigma }_{1,S}& {\sigma }_{1,S}& {\sigma }_{1,S}\end{array}\right],\hfill \\ \hfill & {\varrho }_{2,S}=\left[\begin{array}{cccccccc}0& 1& 0& 0& 1& 1& 0& 1\\ {\varrho }_{1,S}& G_{1,S}^{\alpha }& G_{1,S}^{\alpha }& {\delta }_{1,S}& {\delta }_{1,S}& {\sigma }_{1,S}& {\sigma }_{1,S}& {\varrho }_{1,S}\end{array}\right],\hfill \\ \hfill & {\tau }_{2,S}=\left[\begin{array}{cccccccc}0& 1& 1& 0& 0& 1& 1& 0\\ {\tau }_{1,S}& G_{1,S}^{\alpha }& G_{1,S}^{\alpha }& {\lambda }_{1,S}& {\lambda }_{1,S}& {\sigma }_{1,S}& {\sigma }_{1,S}& {\tau }_{1,S}\end{array}\right],\hfill \\ \hfill & {\nu }_{2,S}=\left[\begin{array}{cccccccc}0& 1& 1& 0& 0& 1& 1& 0\\ {\nu }_{1,S}& G_{1,S}^{\alpha }& G_{1,S}^{\alpha }& {\lambda }_{1,S}& {\lambda }_{1,S}& {\delta }_{1,S}& {\delta }_{1,S}& {\nu }_{1,S}\end{array}\right].\hfill \end{array}$$

Hence, the generator matrix of ${\mathcal{S}}_{3,2}^{\beta }$ is

$$G_{3,2,S}^{\beta }=\left[\begin{array}{ccccccc}1& 0& 1& 1& 1& 0& 0\\ G_{2,S}^{\alpha }& {\lambda }_{2,S}& {\delta }_{2,S}& {\sigma }_{2,S}& {\varrho }_{2,S}& {\tau }_{2,S}& {\nu }_{2,S}\end{array}\right].$$

By using Magma [30], we are aware that ${\mathcal{S}}_{3,2}^{\beta }$ is an $[316,3,178]$ linear code over ${\mathbb{F}}_2$. The result is consistent with Example 2.

Next, we will establish secret sharing schemes with the basis of ${{\mathcal{S}}_{k,h}^{\beta }}^{\perp }$. In this case, there exist 315 participants and 4 minimal access sets. The four minimal access sets are given in the following manner.

$$\begin{array}{cc}\hfill & d_1=(111111111111111111111111111111111111111111111111111111111111111100000\hfill \\ \hfill & 00000000000000000000000000000000000000000001111111111111111111111111111111\hfill \\ \hfill & 11111111111111111111111111111111111111111111111111111111111111111111111111\hfill \\ \hfill & 11111111111111111111111111100000000000000000000000000000000000000000000000\hfill \\ \hfill & 0000000000000000000000000)\hfill \\ \hfill & d_2=(100110011001100110011001100110011001100110011001100110011001100111110\hfill \\ \hfill & 11001100110011001100110011001101111111111110101100110011001100110011001100\hfill \\ \hfill & 11001010101010101001110011001100110011001100110011001001100110011011001100\hfill \\ \hfill & 11001100101010101001100110111011001100110011011111111110011001111011001100\hfill \\ \hfill & 1100110111111111010101011)\hfill \\ \hfill & d_3=(111111110000000000000000111111111111111100000000000000001111111100001\hfill \\ \hfill & 11111111111111111111111111111110000000000001111000000001111111100000000111\hfill \\ \hfill & 11111000000001111111100000000000000001111111111111111000011110000110000000\hfill \\ \hfill & 01111111111110000000011110000111111111111111100000000111111110000111111111\hfill \\ \hfill & 1111111000000001111111100)\hfill \\ \hfill & d_4=(100110010110011001100110100110011001100101100110011001101001100111111\hfill \\ \hfill & 00110011001100110011001100110011111111111110101011001101001100101100110100\hfill \\ \hfill & 11001101010100101001101100110011001101001100110011001110000111100010110011\hfill \\ \hfill & 01001100101011010110000111011100110011001100111111111001100111111100110011\hfill \\ \hfill & 0011001111111110101010111)\hfill \end{array}$$

From Theorem 4, we obtain participants $P_l$, $l\in U=\{3,4,7,24,27,28,31,32,35,36,39,56,59,60,63,113,115,124,127,128,131,140,143,144,147,157,159,162,163,180,183,184,187,188,191,192,195,202,203,209,218,221,222,225,227,228,240,241\}$ is contained in every minimal access set. Meanwhile, the remaining participants, labeled as $P_j$, $0<j\notin U$ must be in two among four minimal access sets.

Next, we demonstrate how to construct the symmetric authentication codes using the complete weight distribution. A systematic authentication code is a quadruple $(\mathcal{U},\mathcal{V},\mathcal{W},\{E_w,w\in \mathcal{W}\})$. Here, $\mathcal{U}$ represents the source state space which is associated with a probability distribution, $\mathcal{V}$ represents the tag space, $\mathcal{W}$ represents the key space, and $E_w:\mathcal{U}\to \mathcal{V}$ is regarded as an encoding rule. For additional insights, consult [31]. We analyze two attack methods, impersonation and substitution attacks. Let $P_I$ represent the maximum success probability for impersonation attack, and $P_S$ for substitution attack. The design of authentication codes should be robust against the most challenging scenarios, requiring that $P_I$ and $P_S$ be minimized. There are two lower bounds provided to guide this optimization process [32]:

$$P_I\ge {\displaystyle \frac{1}{|\mathcal{V}|}},P_S\ge {\displaystyle \frac{1}{|\mathcal{V}|}}.$$

If $P_I=P_S={\displaystyle \frac{1}{|\mathcal{V}|}}$, then the systematic authentication code is called optimal.

Utilizing the technique described in [31], we make use of linear codes to design systematic authentication codes. Denote $\mathcal{C}$ as an $[n,k,d]$ linear code over ${\mathbb{F}}_p$ with p a prime, and $c_j=(c_{j,0},c_{j,1},\cdots ,c_{j,n-1})\in \mathcal{C}$, $0\le j\le p^k-1$. A systematic authentication code is given in the following manner:

$$(\mathcal{U},\mathcal{V},\mathcal{W},\{E_w:w\in \mathcal{W}\})=({\mathbb{Z}}_{p^k},{\mathbb{F}}_p,{\mathbb{Z}}_n\times {\mathbb{F}}_p,\{E_w:w\in \mathcal{W}\}),$$

where $E_w(b)=c_{b,w_1}+w_2$, $b\in \mathcal{U}$, $w=(w_1,w_2)\in \mathcal{W}$. From [31], we know that $P_I={\displaystyle \frac{1}{p}}$ and $P_S=\underset{0\ne b\in \mathcal{C}}{max}\underset{y\in {\mathbb{F}}_p}{max}{\displaystyle \frac{N(b,y)}{n}}$, where $N(b,y)$ represents the number of times that y occurs in the codeword b.

Since torsion codes ${\mathcal{S}}_{k,h}^{\alpha }$ and ${\mathcal{S}}_{k,h}^{\beta }$ are linear codes over ${\mathbb{F}}_2$, then by the above method and Theorems 1 and 2, systematic authentication codes can be built.

Theorem 5. 

Denote ${\mathcal{S}}_{k,h}^{\alpha }$ as a linear code of length $2^{3k}-2^{3h}$ over ${\mathbb{F}}_2$, where $k\ge 2$, $1\le h\le k-1$. Then

$$P_I={\displaystyle \frac{1}{2}},P_S={\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2(2^{3k-3h}-1)}},|\mathcal{U}|=2^k,|\mathcal{V}|=2,|\mathcal{W}|=2^{3k+1}-2^{3h+1}.$$

It is clear that $\underset{k\to +\infty }{lim}{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2(2^{3k-3h}-1)}}={\displaystyle \frac{1}{2}}$. Therefore, as k increases significantly, the authentication codes in Theorem 5 are asymptotically optimal.

Theorem 6. 

Denote ${\mathcal{S}}_{k,h}^{\beta }$ as a linear code of length ${(2^k-1)}^3-{(2^h-1)}^3$ over ${\mathbb{F}}_2$, where $k\ge 2$, $1\le h\le k-1$. Then

$$P_I={\displaystyle \frac{1}{2}},P_S={\displaystyle \frac{1}{2}}·{\displaystyle \frac{2^{3k-1}-2^{2k}+2^{k-1}}{2^{3k-1}-3·2^{2k-1}+3·2^{k-1}-2^{3h-1}+3·2^{2h-1}-3·2^{h-1}}},$$

$$|\mathcal{U}|=2^k,|\mathcal{V}|=2,|\mathcal{W}|=2({(2^k-1)}^3-{(2^h-1)}^3).$$

Since

$$\begin{array}{cc}\hfill & \underset{k\to +\infty }{lim}{\displaystyle \frac{1}{2}}·{\displaystyle \frac{2^{3k-1}-2^{2k}+2^{k-1}}{2^{3k-1}-3·2^{2k-1}+3·2^{k-1}-2^{3h-1}+3·2^{2h-1}-3·2^{h-1}}}\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}\underset{k\to +\infty }{lim}{\displaystyle \frac{2^{3k-1}-2^{2k}+2^{k-1}}{2^{3k-1}-3·2^{2k-1}+3·2^{k-1}-2^{3h-1}+3·2^{2h-1}-3·2^{h-1}}}\hfill \\ \hfill & ={\displaystyle \frac{1}{2}},\hfill \end{array}$$

then as k increases significantly, the authentication codes in Theorem 6 are asymptotically optimal.

5. Conclusions

In this paper, we explored the algebraic structure of the torsion codes of MacDonald codes of types $\alpha$ and $\beta$ over $R={\mathbb{F}}_2+w_1{\mathbb{F}}_2+w_2{\mathbb{F}}_2$, where $w_1^2=w_1$, and $w_2^2=w_2$, $w_1{w}_2=w_2{w}_1=0$. We obtain the complete weight enumerator of ${\mathcal{S}}_{k,h}^{\alpha }$ and ${\mathcal{S}}_{k,h}^{\beta }$ and their applications. Furthermore, if $w_1{w}_2=w_2{w}_1\ne 0$, then $R={\mathbb{F}}_2+w_1{\mathbb{F}}_2+w_2{\mathbb{F}}_2+w_1{w}_2{\mathbb{F}}_2$. Studying the torsion codes of MacDonald codes of types $\alpha$ and $\beta$ and their applications over ${\mathbb{F}}_2+w_1{\mathbb{F}}_2+w_2{\mathbb{F}}_2+w_1{w}_2{\mathbb{F}}_2$ might be a job full of interest. Applying complete weight enumerator to the study of quantum codes and nonlinear codes will also be an interesting job.