Group codes over symmetric groups

1.   Introduction

Group codes, a class of important linear codes, play a vital role in error correction coding. A linear code $\mathcal{C}$ is called a group code if it is just a one-sided (left or right) ideal in a group algebra $R[G]$, where $R$ is a commutative ring and $G$ is a finite group. If $G$ is abelian, then $\mathcal{C}$ is an abelian code.

A brief survey on group codes of some recent results is provided as follows. Polcino Milies et al. ^[13] calculated the minimum distances and the dimensions of all cyclic codes of length $p^{n}$ over a finite field $\Bbb F_q$, when $p$ is an odd prime and $\Bbb F_q$ is a finite field with $q$ elements, assuming that $\bar{q}$ generates the group of invertible elements of the residue ring module $p^n$, denoted by $\mathbb{Z}_{p^n}$. Jitman et al. ^[11] gave a characterization and an enumeration of Euclidean self-dual and Euclidean self-orthogonal abelian codes in a principal ideal group algebra. Choosuwan et al. ^[5] gave the complete enumeration of self-dual abelian codes in nonprincipal ideal group algebras $\Bbb F_{2^k}[A\times \mathbb{Z}_2 \times \mathbb{Z}_{2^s}]$ with respect to both the Euclidean and Hermitian inner products, where $k$ and $s$ are positives and $A$ is an abelian group of odd order. In 2017, Boripan et al. ^[1] studied a family of abelian codes with complementary dual in a group algebra $\Bbb F_{p^v}[G]$ in the two cases of Euclidean and Hermitian inner products, where $p$ is a prime, $v$ is a positive integer, and $G$ is an arbitrary finite abelian group. Cao et al. ^[3,4] proved that any left $D_{2n}$-code (left ideal of the group algebra $\Bbb F_q[D_{2n}]$) is a direct sum of concatenated codes with inner codes $A_{i}$ and outer codes $C_{i}$, where $A_{i}$ is a minimal self-reciprocal cyclic code over $\Bbb F_q$ of length $n$ and $C_{i}$ is a skew cyclic code of length 2 over an extension field or principal ideal ring of $\Bbb F_q$.

Determining the parameters of linear codes is important, but it is difficult to determine the parameters of group codes in group algebras $\Bbb F_q[G]$ through the structure of $\Bbb F_q[G]$. For instance, Brochero Martínes ^[2] showed explicitly all central irreducible idempotents and their Wedderburn decomposition of the dihedral group algebra $\Bbb F_q[D_{2n}]$, in the case when every divisor of $n$ divides $q-1$. Brochero Martínez et al. ^[12] determined an explicit expression for the primitive idempotents of $\Bbb F_q[G]$, where $\Bbb F_q$ is a finite field, $G$ is a finite cyclic group of order $p^{k}$ and $p$ is an odd prime with $\gcd(q, p) = 1$. Based on the idea, Gao et al. ^[7] described and counted all linear complementary dual (LCD) codes and self-orthogonal codes in $\Bbb F_q[D_{2n, r}]$, where $[D_{2n, r}]$ is a generalized dihedral group. Gao et al. ^[6] obtained the precise descriptions and enumerations of linear complementary dual (LCD) codes and self-orthogonal codes in the generalized quaternion group algebras $\Bbb F_q[Q_{4n}]$. It is a pity that the authors cannot give the parameters of group codes.

To address this issue, we can use the character label of corresponding groups to determine the idempotents of group algebras. Let $S_n$ be a symmetric group of order $n!$. In this paper, we propose group codes in symmetric group algebras $\Bbb F_q S_n$ with $q > 3$ and $n = 3, 4$. We compute the unique (linear and nonlinear) idempotents of $\Bbb F_q S_n$ corresponding to the characters of symmetric groups and use the results to characterize the minimum distances and dimensions of group codes. Furthermore, we construct MDS group codes and almost MDS group codes in $\Bbb F_q S_3$ and $\Bbb F_q S_4$.

The paper is organized as follows: Section 2 provides a review of some properties of group algebras and other preliminaries, while Section 3 investigates group codes in symmetric group algebras $\Bbb F_q S_n$ with $q > 3$ and $n = 3, 4$, and obtains the parameters of all the above group codes.

2.   Preliminaries

Let $\Bbb F_q[G]$ be a group algebra, where $\Bbb F_q$ is a finite field and $G$ is a finite group. In fact, the group algebra $\Bbb F_q[G]$ is a vector space over $\Bbb F_q$ with basis $G$, and it has scalar, additive, and multiplicative operator as follows: for $c, a_g, b_g \in \Bbb F_q$ and $g \in G$,

Then $\Bbb F_q[G]$ is an associative $\Bbb F_q$-algebra with the identity $1 = 1_ {\Bbb F_q}1_G$, where $1_ {\Bbb F_q}$ and $1_G$ are the identity elements of $\Bbb F_q$ and $G$, respectively. Readers are referred to ^[14,16] for more details on group ring or group algebra.

Lemma 2.1. (Maschke's Theorem ^[14]) Let $R$ be a ring and $G$ be a group. Then the group ring $R[G]$ is semisimple if and only if the following conditions hold.

(i) $R$ is a semisimple ring.

(ii) $G$ is finite.

(iii) $|G|$ is invertible in $R$.

By Lemma 2.1, it is easy to verify that $\Bbb F_q[G]$ is semisimple if and only if $G$ is a finite group and $char(\Bbb F_q)\nmid |G|$. Let $\Bbb F_q[G]$ be a semisimple group algebra. Then $\Bbb F_q[G]$ can be decomposed into a direct sum $\Bbb F_q[G] = \bigoplus \limits_{i\in \, \Omega}\Bbb F_q[G]e_i$, where $\Bbb F_q[G]e_i$ is the minimal ideal generated by the idempotent $e_i, i\in \Omega$ and $\Omega$ is the index set (see ^[9]). If $I$ is any ideal of $\Bbb F_q[G]$, then $I$ can be expressed as a direct sum of some minimal ideal $\Bbb F_q[G]e_i$ of $\Bbb F_q[G]$, i.e., $I = \bigoplus \limits_{i\in \, \Omega_1\subseteq \Omega}\Bbb F_q[G]e_i$. Let $I$ be an ideal generated by a subset $\Omega_1$ and $\Lambda = \Omega \setminus \Omega_1$. Then $I\doteq I_{\Lambda} = \{\alpha \in \Bbb F_q[G]: \alpha e_j = 0, \ \mbox{for all}\ e_j \in \Lambda\}$.

Suppose that $\alpha = \sum\limits_{g\in\, G}a_gg\in \Bbb F_q[G]$. Then $wt(\alpha) = |\{a_g: a_g\neq 0\}|$ is called the Hamming weight of $\alpha$ (^[19]). Let $I_{\Lambda}$ be a group code in $\Bbb F_q[G]$. Its length $n$ is the order of $G$ in group algebra $\Bbb F_q[G]$. Its dimension $k$ is the dimension of $I_{\Lambda}$ as a subspace over $\Bbb F_q$. Its minimal distance $d$ is defined as $d = \min\{wt(\alpha): \alpha e_i = 0 \ \mbox{for nonzero element}\ \alpha \ \mbox{and for all} \ e_i \in \Lambda\}$. Thus the group code $I_{\Lambda}$ is called an $[n, k, d]$ code. Moreover, a linear $[n, k, d]$ group code over $\Bbb F_q$ with $d = n-k+1$ is called a maximum distance separable (MDS) group code. And, a linear $[n, k, d]$ group code over $\Bbb F_q$ with $d = n-k$ is called an almost MDS (AMDS) group code. MDS codes and AMDS codes are considered to be an attractive solution for information storage as they operate at the optimal storage versus reliability trade-off (see ^{[8,15,17,18,20,21]}).

Based on the following lemma, we can give the unique idempotents of $\Bbb F_q[G]$.

Lemma 2.2. ^[10] If $\chi$ is a character of $\Bbb F_qG$-module, then the idempotents corresponding to the character $\chi$ are given by

3.   Group codes in $\Bbb F_qS_n$

3.1.   Group codes in $\Bbb F_qS_3$

In the subsection, we will consider the group codes on $S_3$, where $S_3$ is a nonabelian group with the smallest order. Let

It is well-known that $S_3$ is all the permutations on three elements 1, 2, 3. In this sense, set $a = (123), b = (12)$, $S_3$ can be given by $\{(1), (123), (132), (12), (23), (13)\}$. For convenience, we denote $g_i\; (1\leq i\leq 6)$ as the $i$-th element of $S_3$ in the above two sets.

In addition, $S_3$ has three conjugacy classes as follows:

Since the commutator group of $S_3$ is $S'_3 = \{1, a, a^2\}$, we have $|S_3/S'_3| = 2$. As a consequence of this, the group $S_3$ has two linear characters $\chi_1, \chi_2$ and one nonlinear character $\chi_3$. The character table of $S_3$ is as follows (see Table 1).

By Lemma 2.2, the unique idempotents of $\Bbb F_qS_3$ are given in the following theorem.

Theorem 3.1. There are three idempotents in $\Bbb F_qS_3$ as follows:

where $\overline{C}_i = \sum\limits_{g\in C_i}g, i = 2, 3.$

In order to construct the group codes of $\Bbb F_qS_3$, we need the product between idempotents $e_1, e_2, f_1$ and an arbitrary element of $\Bbb F_qS_3$. For any element $\alpha = \sum\limits_{i = 1}^{6}a_ig_i\in \Bbb F_qS_3$, where $a_i\in \Bbb F_q, i = 1, 2, \ldots, 6$, we have

The following theorems give us the parameters of group codes in $\Bbb F_qS_3$.

Theorem 3.2. Let $e_1, e_2$ and $f_1$ be idempotents in $\Bbb F_qS_3$. Then

(1) $I_{\{e_1\}}$ is a [6,5,2] group code;

(2) $I_{\{e_2\}}$ is a [6,5,2] group code;

(3) $I_{\{f_1\}}$ is a [6,5,3] group code.

Proof. (1) Clearly, $I_{\{e_1\}} = \{\alpha \in \Bbb F_qS_3|\alpha e_1 = 0\}.$ Firstly, we will give the dimension of group code $I_{\{e_1\}}$. For any $\alpha = \sum\limits_{i = 1}^{6}a_ig_i\in I_{\{e_1\}}$, we have $\alpha e_1 = 0$. By (3.1), we obtain $\sum\limits_{i = 1}^{6}a_i = 0$. Then $\dim(I_{\{e_1\}}) = 5$. Secondly, we will compute the minimal distance of group code $I_{\{e_1\}}$. For some $\alpha = kg_i, 1\leq i \leq 6$, if $k \neq 0$, then $\alpha e_1 \neq 0$, i.e., $\alpha = kg_i \notin I_{\{e_1\}}$. So $d(I_{\{e_1\}})\geq 2$. Set $\alpha = g_1-g_2$. Since $\alpha e_1 = (g_1-g_2)e_1 = 0$, we get $\alpha = g_1-g_2 \in I_{\{e_1\}}$. Hence, $d(I_{\{e_1\}}) = 2$.

(2) Clearly, $I_{\{e_2\}} = \{\alpha \in \Bbb F_qS_3|\alpha e_2 = 0\}.$ Firstly, we will give the dimension of group code $I_{\{e_2\}}$. For any $\alpha = \sum\limits_{i = 1}^{6}a_ig_i\in I_{\{e_2\}}$, we have $\alpha e_2 = 0$. By (3.2), we obtain $\sum\limits_{i = 1}^{3}a_i-\sum\limits_{i = 4}^{6}a_i = 0$. Then $\dim(I_{\{e_2\}}) = 5$. Secondly, we will compute the minimal distance of group code $I_{\{e_2\}}$. For some $\alpha = kg_i, 1\leq i \leq 6$, if $k \neq 0$, then $\alpha e_2 \neq 0$, i.e., $\alpha = kg_i \notin I_{\{e_2\}}$. So $d(I_{\{e_2\}})\geq 2$. Set $\alpha = g_1+g_4$. Since $\alpha e_2 = (g_1+g_4)e_2 = 0$, we get $\alpha = g_1+g_4 \in I_{\{e_2\}}$. Hence, $d(I_{\{e_2\}}) = 2$.

(3) Clearly, $I_{\{f_1\}} = \{\alpha \in \Bbb F_qS_3|\alpha f_1 = 0\}.$ Firstly, we will give the dimension of group code $I_{\{f_1\}}$. For any $\alpha = \sum\limits_{i = 1}^{6}a_ig_i\in I_{\{f_1\}}$, we have $\alpha f_1 = 0$. By (3.3), we obtain

Since the rank of the coefficient matrix of the above equation is 4, we know that $\dim(I_{\{f_1\}}) = 2$. Secondly, we will compute the minimal distance of group code $I_{\{f_1\}}$. For some $\alpha = k_ig_i+k_jg_j, 1\leq i, j \leq 6$, if $k_i, k_j \neq 0$, then $\alpha f_1 \neq 0$, i.e., $\alpha = k_ig_i+k_jg_j \notin I_{\{f_1\}}$. So $d(I_{\{f_1\}})\geq 3$. Set $\alpha = g_1+g_2+g_3$. Since $\alpha f_1 = (g_1+g_2+g_3)f_1 = 0$, we get $\alpha = g_1+g_2+g_3 \in I_{\{f_1\}}$. Hence, $d(I_{\{f_1\}}) = 3$.

This completes the proof. □

Theorem 3.3. Let $e_1, e_2$ and $f_1$ be idempotents in $\Bbb F_qS_3$. Then

(1) $I_{\{e_1, e_2\}}$ is a [6,4,2] group code;

(2) $I_{\{e_1, f_1\}}$ is a [6,1,6] group code;

(3) $I_{\{e_2, f_1\}}$ is a [6,1,6] group code.

Proof. (1) Clearly, $I_{\{e_1, e_2\}} = \{\alpha \in \Bbb F_qS_3|\alpha e_1 = 0, \alpha e_2 = 0\}.$ Firstly, we will give the dimension of group code $I_{\{e_1, e_2\}}$. For any $\alpha = \sum\limits_{i = 1}^{6}a_ig_i\in I_{\{e_1, e_2\}}$, we have $\alpha e_1 = 0$ and $\alpha e_2 = 0$. From (3.1) and (3.2), we obtain

Since the rank of the coefficient matrix of the above equation is 2, we know that $\dim(I_{\{e_1, e_2\}}) = 4$. Secondly, we will compute the minimal distance of group code $I_{\{e_1, e_2\}}$. For some $\alpha = kg_i, 1\leq i \leq 6$, if $k \neq 0$, then $\alpha e_1 \neq 0$, i.e., $\alpha = kg_i \notin I_{\{e_1, e_2\}}$. So $d(I_{\{e_1, e_2\}})\geq 2$. Set $\alpha = g_1-g_3$. Since $\alpha e_1 = (g_1-g_3)e_1 = 0$ and $\alpha e_2 = (g_1-g_3)e_2 = 0$, we get $\alpha = g_1-g_3 \in I_{\{e_1, e_2\}}$. Hence, $d(I_{\{e_1, e_2\}}) = 2$.

(2) Clearly, $I_{\{e_1, f_1\}} = \{\alpha \in \Bbb F_qS_3|\alpha e_1 = 0, \alpha f_1 = 0\}.$ Firstly, we will give the dimension of group code $I_{\{e_1, f_1\}}$. For any $\alpha = \sum\limits_{i = 1}^{6}a_ig_i\in I_{\{e_1, f_1\}}$, we have $\alpha e_1 = 0$ and $\alpha f_1 = 0$. From (3.1) and (3.3), we obtain

Since the rank of the coefficient matrix of the above equations is 5, we know that $\dim(I_{\{e_1, f_1\}}) = 1$. Secondly, we will compute the minimal distance of group code $I_{\{e_1, f_1\}}$. If we take $\alpha = g_1+g_2+g_3$, we have $\alpha f_1 = 0$ and $\alpha e_1 \neq 0$, i.e., $\alpha = g_1+g_2+g_3 \notin I_{\{e_1, f_1\}}$. So $d(I_{\{e_1, e_2\}})\geq 3$. Set $\alpha = g_1+g_2+g_3-g_4-g_5-g_6$. Since $\alpha e_1 = (g_1+g_2+g_3-g_4-g_5-g_6)e_1 = 0$ and $\alpha e_2 = (g_1+g_2+g_3-g_4-g_5-g_6)f_1 = 0$, we get $\alpha = g_1+g_2+g_3-g_4-g_5-g_6 \in I_{\{e_1, f_1\}}$. Hence, $d(I_{\{e_1, f_1\}}) = 6$.

(3) The result can be obtained by a similar proof of (2).

This completes the proof. □ We can get the following results based on Theorems 3.2 and 3.3.

Remark 3.4. Let $e_1, e_2$ and $f_1$ be idempotents in $\Bbb F_qS_3$.

(1) $I_{\{e_i\}}, I_{\{e_i, f_1\}}, i = 1, 2$ are MDS group codes;

(2) $I_{\{e_1, e_2\}}$ is an AMDS group codes.

3.2.   Group codes in $\Bbb F_qS_4$

In the subsection, we will consider the group codes on $S_4$, where $S_4$ is a nonabelian group of order 24. Let

It is well-known that $S_4$ is all the permutations on four elements 1, 2, 3, 4. In this sense, set $a = (1234), b = (12)$, $S_4$ can be given by {(1), (1234), (13)(24), (1432), (12), (234), (1324), (143), (134), (1243), (142), (23), (1423), (132), (34), (124), (243), (14), (123), (1342), (14)(23), (13), (12)(34), (24)}. For convenience, we denote $g_i$ $(1\leq i\leq 24)$ as the $i$-th element of $S_4$ in the above two sets.

In addition, $S_4$ has five conjugacy classes as follows:

Since the commutator group of $S_4$ is $S'_4 = A_4$, we have $|S_4/S'_4| = 2$. As a consequence of this, the group $S_4$ has two linear characters $\chi_1, \chi_2$ and three nonlinear characters $\chi_3$, $\chi_4$, $\chi_5$. The character table of $S_4$ is as follows (see Table 2).

By Lemma 2.2, the unique idempotents of $\Bbb F_qS_4$ are given in the following theorem.

Theorem 3.5. There are five idempotents in $\Bbb F_qS_4$ as follows:

where $\overline{C}_i = \sum\limits_{g\in C_i}g, i = 2, 3, 4, 5.$

In order to construct the group codes of $\Bbb F_qS_4$, we need the product between idempotents $e_1, e_2, f_1, f_2, f_3$ and an arbitrary element of $\Bbb F_qS_4$. For any element $\alpha = \sum\limits_{i = 1}^{24}a_ig_i\in \Bbb F_qS_4$, where $a_i\in \Bbb F_q, i = 1, 2, \ldots, 24$, we have

$\alpha f_1 = \frac{1}{24}[(2a_{1}+2a_{3}+2a_{21}+2a_{23}-a_{6}-a_{8}-a_{9}-a_{11}-a_{14}-a_{16}-a_{17}-a_{19}) (g_{1}+g_{3}+g_{21}+g_{23}) +(2a_{2}+2a_{4}+2a_{22}+2a_{24}-a_{5}$-$a_{7}-a_{10}-a_{12}-a_{13}-a_{15}-a_{18}-a_{20}) (g_{2}+g_{4}+g_{22}+g_{24}) +(2a_{5}+2a_{7}+2a_{13}+2a_{15}$-$a_{2}-a_{4}-a_{10}-a_{12}-a_{18}-a_{20}-a_{22}-a_{24}) (g_{5}+g_{7}+g_{13}+g_{15}) +(2a_{6}+2a_{8}+2a_{14}+2a_{16}-a_{1}-a_{3}-a_{9}-a_{11}-a_{17}-a_{19}-a_{21}-a_{23}) (g_{6}+g_{8}+g_{14}+g_{16}) +(2a_{9}+2a_{11}+2a_{17}+2a_{19}-a_{1}-a_{3}$-$a_{6}-a_{8}-a_{14}-a_{16}-a_{21}-a_{23}) (g_{9}+g_{11}+g_{17}+g_{19}) +(2a_{10}+2a_{12}+2a_{18}+2a_{20}-a_{2}-a_{4}-a_{5}-a_{7}-a_{13}-a_{15}-a_{22}-a_{24}) (g_{10}+g_{12}+g_{18}+g_{20})$.

$\alpha f_2 = \frac{1}{24}[(3a_{1}+a_{5}+a_{12}+a_{15}+a_{18}+a_{22}+a_{24}-a_{3}-a_{21}-a_{23}-a_{2}-a_{4}-a_{7}-a_{10}-a_{13}-a_{20})g_{1} +(3a_{2}+a_{6}+a_{9}+a_{16}+a_{19}+a_{21}+a_{23}$-$a_{4}-a_{22}-a_{24}-a_{1}-a_{3}-a_{8}-a_{11}-a_{14}-a_{17})g_{2} +(3a_{3}+a_{7}+a_{10}+a_{13}+a_{20}+a_{22}+a_{24}-a_{1}-a_{21}$-$a_{23}-a_{2}-a_{4}-a_{5}-a_{12}-a_{15}-a_{18})g_{3} +(3a_{4}+a_{8}+a_{11}+a_{14}+a_{17}+a_{21}+a_{23}-a_{2}-a_{22}$-$a_{24}-a_{1}-a_{3}-a_{6}-a_{9}-a_{16}-a_{19})g_{4} +(3a_{5}+a_{1}+a_{11}+a_{14}+a_{16}+a_{19}+a_{23}$-$a_{7}-a_{13}-a_{15}-a_{2}-a_{6}-a_{8}-a_{9}-a_{17}-a_{21})g_{5} +(3a_{6}+a_{2}+a_{12}+a_{13}+a_{15}+a_{20}+a_{24}-a_{8}-a_{14}$-$a_{16}-a_{3}-a_{5}-a_{7}-a_{10}-a_{18}-a_{22})g_{6} +(3a_{7}+a_{3}+a_{9}+a_{14}+a_{16}+a_{17}+a_{21}-a_{5}-a_{13}-a_{15}-a_{1}-a_{6}-a_{8}-a_{11}-a_{19}-a_{23})g_{7} +(3a_{8}+a_{4}+a_{10}+a_{13}+a_{15}+a_{18}+a_{22}-a_{6}$-$a_{14}-a_{16}-a_{2}-a_{5}-a_{7}-a_{12}-a_{20}-a_{24})g_{8} +(3a_{9}+a_{2}+a_{7}+a_{15}+a_{18}+a_{20}+a_{22}-a_{11}-a_{17}-a_{19}-a_{4}-a_{5}-a_{10}-a_{12}-a_{13}-a_{24})g_{9} +(3a_{10}+a_{3}+a_{8}+a_{16}+a_{17}+a_{19}+a_{23}-a_{12}$-$a_{18}-a_{20}-a_{1}-a_{6}-a_{9}-a_{11}-a_{14}-a_{21})g_{10} +(3a_{11}+a_{4}+a_{5}+a_{13}+a_{18}+a_{20}+a_{24}-a_{9}-a_{17}-a_{19}-a_{2}-a_{7}-a_{10}-a_{12}-a_{15}-a_{22})g_{11} +(3a_{12}+a_{1}+a_{6}+a_{14}+a_{17}+a_{19}+a_{21}-a_{10}-a_{18}-a_{20}-a_{3}-a_{8}-a_{9}-a_{11}-a_{16}-a_{23})g_{12} +(3a_{13}+a_{3}+a_{6}+a_{8}+a_{11}+a_{19}$+$a_{21}-a_{5}-a_{7}-a_{15}-a_{1}-a_{9}-a_{14}-a_{16}-a_{17}-a_{23})g_{13} +(3a_{14}+a_{4}+a_{5}+a_{7}+a_{12}+a_{20}+a_{22}-a_{6}-a_{8}-a_{16}$-$a_{2}-a_{10}-a_{13}-a_{15}-a_{18}-a_{24})g_{14} +(3a_{15}+a_{1}+a_{6}+a_{8}+a_{9}+a_{17}+a_{23}-a_{5}-a_{7}-a_{13}-a_{3}-a_{11}-a_{14}-a_{16}-a_{19}-a_{21})g_{15} +(3a_{16}+a_{2}+a_{5}+a_{7}+a_{10}+a_{18}+a_{24}-a_{6}-a_{8}-a_{14}$-$a_{4}-a_{12}-a_{13}-a_{15}-a_{20}-a_{22})g_{16} +(3a_{17}+a_{4}+a_{7}+a_{10}+a_{12}+a_{15}+a_{24}-a_{9}-a_{11}-a_{19}-a_{2}-a_{5}-a_{13}-a_{18}-a_{20}-a_{22})g_{17} +(3a_{18}+a_{1}+a_{8}+a_{9}+a_{11}+a_{16}+a_{21}-a_{10}-a_{12}-a_{20}-a_{3}-a_{6}-a_{14}-a_{17}-a_{19}-a_{23})g_{18} +(3a_{19}+a_{2}+a_{5}+a_{10}+a_{12}+a_{13}+a_{22}$-$a_{9}-a_{11}$-$a_{17}-a_{4}-a_{7}-a_{15}-a_{18}-a_{20}-a_{24})g_{19} +(3a_{20}+a_{3}+a_{6}+a_{9}$+$a_{11}+a_{14}+a_{23}-a_{10}-a_{12}-a_{18}-a_{1}-a_{8}-a_{16}-a_{17}-a_{19}-a_{21})g_{20} +(3a_{21}+a_{2}+a_{4}+a_{7}+a_{12}+a_{13}+a_{18}-a_{1}-a_{3}-a_{23}-a_{5}$-$a_{10}-a_{15}-a_{20}-a_{22}-a_{24})g_{21} +(3a_{22}+a_{1}+a_{3}+a_{8}+a_{9}+a_{14}$+$a_{19}-a_{2}-a_{4}-a_{24}-a_{6}-a_{11}-a_{16}-a_{17}-a_{21}-a_{23})g_{22} +(3a_{23}+a_{2}+a_{4}+a_{5}+a_{10}+a_{15}+a_{20}-a_{1}-a_{3}-a_{21}-a_{7}-a_{12}-a_{13}-a_{18}-a_{22}-a_{24})g_{23} +(3a_{24}+a_{1}+a_{3}+a_{6}+a_{11}+a_{16}+a_{17}-a_{2}-a_{4}-a_{22}-a_{8}-a_{9}-a_{14}-a_{19}-a_{21}-a_{23})g_{24}]$.

$\alpha f_3 = \frac{1}{24}[(3a_{1}-a_{3}-a_{5}-a_{12}-a_{15}-a_{18}-a_{21}-a_{22}-a_{23}-a_{24}+a_{2}+a_{4}+a_{7}+a_{10}+a_{13}+a_{20})g_{1} +(3a_{2}-a_{4}-a_{6}-a_{9}-a_{16}-a_{19}-a_{21}-a_{22}-a_{23}-a_{24}+a_{1}+a_{3}+a_{8}+a_{11}+a_{14}+a_{17})g_{2} +(3a_{3}-a_{1}-a_{7}-a_{10}-a_{13}-a_{20}$-$a_{21}-a_{22}-a_{23}-a_{24}+a_{2}+a_{4}+a_{5}+a_{12}+a_{15}+a_{18})g_{3} +(3a_{4}-a_{2}-a_{8}-a_{11}-a_{14}-a_{17}-a_{21}$-$a_{22}-a_{23}-a_{24}+a_{1}+a_{3}+a_{6}+a_{9}+a_{16}+a_{19})g_{4} +(3a_{5}-a_{1}-a_{7}-a_{11}-a_{13}-a_{14}$-$a_{15}-a_{16}-a_{19}-a_{23}+a_{2}+a_{6}+a_{8}+a_{9}+a_{17}+a_{21})g_{5} +(3a_{6}-a_{2}-a_{8}-a_{12}-a_{13}-a_{14}-a_{15}-a_{16}$-$a_{20}-a_{24}+a_{3}+a_{5}+a_{7}+a_{10}+a_{18}+a_{22})g_{6} +(3a_{7}-a_{3}-a_{5}-a_{9}-a_{13}-a_{14}-a_{15}$-$a_{16}-a_{17}-a_{21}+a_{1}+a_{6}+a_{8}+a_{11}+a_{19}+a_{23})g_{7} +(3a_{8}-a_{4}-a_{6}-a_{10}-a_{13}-a_{14}-a_{15}-a_{16}-a_{18}-a_{22}$+$a_{2}+a_{5}+a_{7}+a_{12}+a_{20}+a_{24})g_{8} +(3a_{9}-a_{2}-a_{7}$-$a_{11}-a_{15}-a_{17}-a_{18}$-$a_{19}-a_{20}-a_{22}+a_{4}+a_{5}+a_{10}+a_{12}+a_{13}+a_{24})g_{9} +(3a_{10}-a_{3}-a_{8}-a_{12}-a_{16}-a_{17}-a_{18}-a_{19}-a_{20}-a_{23}+a_{1}+a_{6}+a_{9}+a_{11}+a_{14}+a_{21})g_{10} +(3a_{11}-a_{4}-a_{5}-a_{9}-a_{13}-a_{17}-a_{18}-a_{19}$-$a_{20}-a_{24}+a_{2}+a_{7}+a_{10}+a_{12}+a_{15}+a_{22})g_{11} +(3a_{12}-a_{1}-a_{6}-a_{10}-a_{14}-a_{17}-a_{18}-a_{19}-a_{20}-a_{21}+a_{3}+a_{8}+a_{9}+a_{11}+a_{16}+a_{23})g_{12} +(3a_{13}-a_{3}-a_{5}-a_{6}$-$a_{7}-a_{8}-a_{11}-a_{15}-a_{19}-a_{21}+a_{1}+a_{9}+a_{14}+a_{16}+a_{17}+a_{23})g_{13} +(3a_{14}-a_{4}-a_{5}-a_{6}-a_{7}-a_{8}-a_{12}-a_{16}-a_{20}-a_{22}$+$a_{2}+a_{10}+a_{13}+a_{15}+a_{18}+a_{24})g_{14} +(3a_{15}-a_{1}-a_{5}-a_{6}-a_{7}-a_{8}-a_{9}$-$a_{13}-a_{17}-a_{23}+a_{3}+a_{11}+a_{14}+a_{16}+a_{19}+a_{21})g_{15} +(3a_{16}-a_{2}-a_{5}-a_{6}-a_{7}-a_{8}-a_{10}-a_{14}-a_{18}-a_{24}+a_{4}+a_{12}+a_{13}+a_{15}+a_{20}+a_{22})g_{16} +(3a_{17}-a_{4}-a_{7}-a_{9}-a_{10}$-$a_{11}-a_{12}-a_{15}-a_{19}-a_{24}+a_{2}+a_{5}+a_{13}+a_{18}+a_{20}+a_{22})g_{17} +(3a_{18}-a_{1}-a_{8}-a_{9}-a_{10}-a_{11}-a_{12}-a_{16}-a_{20}-a_{21}+a_{3}+a_{6}+a_{14}+a_{17}+a_{19}+a_{23})g_{18} +(3a_{19}-a_{2}-a_{5}-a_{9}$-$a_{10}-a_{11}-a_{12}$-$a_{13}-a_{17}-a_{22}+a_{4}+a_{7}+a_{15}+a_{18}+a_{20}+a_{24})g_{19} +(3a_{20}-a_{3}-a_{6}-a_{9}-a_{10}-a_{11}-a_{12}-a_{14}-a_{18}-a_{23}+a_{1}+a_{8}+a_{16}+a_{17}+a_{19}+a_{21})g_{20} +(3a_{21}-a_{1}-a_{2}$-$a_{3}-a_{4}-a_{7}-a_{12}-a_{13}-a_{18}-a_{23}+a_{5}+a_{10}$+$a_{15}+a_{20}+a_{22}+a_{24})g_{21} +(3a_{22}-a_{1}-a_{2}-a_{3}-a_{4}-a_{8}$-$a_{9}-a_{14}-a_{19}-a_{24}+a_{6}+a_{11}+a_{16}+a_{17}+a_{21}+a_{23})g_{22} +(3a_{23}-a_{1}-a_{2}-a_{3}-a_{4}-a_{5}-a_{10}-a_{15}$-$a_{20}-a_{21}+a_{7}+a_{12}+a_{13}+a_{18}+a_{22}+a_{24})g_{23} +(3a_{24}-a_{1}-a_{2}-a_{3}-a_{4}-a_{6}-a_{11}-a_{16}-a_{17}-a_{22}+a_{8}+a_{9}+a_{14}+a_{19}+a_{21}+a_{23})g_{24}].$

The following theorems give us the parameters of group codes in $\Bbb F_qS_4$.

Theorem 3.6. Let $e_i, i = 1, 2$ and $f_j, j = 1, 2, 3$ be idempotents in $\Bbb F_qS_4$. Then

(1) $I_{\{e_i\}}$ is a [24,23,2] group code, for $i = 1, 2$;

(2) $I_{\{f_1\}}$ is a [24,20,2] group code;

(3) $I_{\{f_j\}}$ is a [24,15,4] group code, for $j = 2, 3$.

Proof. (1) Clearly, $I_{\{e_i\}} = \{\alpha \in \Bbb F_qS_4|\alpha e_i = 0\},$ for $i = 1, 2$. Firstly, we will give the dimension of group code $I_{\{e_i\}}, i = 1, 2$. For any $\alpha = \sum\limits_{i = 1}^{24}a_ig_i\in I_{\{e_i\}}$, we have $\alpha e_i = 0$. By (3.4), we obtain the following equation:

By (3.5), we obtain the following equation:

Then, for $i = 1, 2$, we have $\dim(I_{\{e_i\}}) = 5$. Secondly, we will compute the minimal distance of group code $I_{\{e_i\}}$. For some $\alpha = kg_i, 1\leq i \leq 24$, if $k \neq 0$, then $\alpha e_i \neq 0$, i.e., $\alpha = kg_i \notin I_{\{e_i\}}$. So $d(I_{\{e_i\}})\geq 2$. Set $\alpha = g_1-g_3$. Since $\alpha e_i = (g_1-g_3)e_i = 0$, we get $\alpha = g_1-g_3 \in I_{\{e_i\}}$. Hence, $d(I_{\{e_i\}}) = 2$.

(2) Clearly, $I_{\{f_1\}} = \{\alpha \in \Bbb F_qS_4|\alpha f_1 = 0\}$. Firstly, we will give the dimension of group code $I_{\{f_1\}}$. For any $\alpha = \sum\limits_{i = 1}^{24}a_ig_i\in I_{\{f_1\}}$, we have $\alpha f_1 = 0$. Then, we obtain the following equations:

Since the rank of the coefficient matrix of the above equations is 4, we know that $\dim(I_{\{f_1\}}) = 20$. Secondly, we will compute the minimal distance of group code $I_{\{f_1\}}$. For some $\alpha = kg_i, 1\leq i \leq 24$, if $k \neq 0$, then $\alpha f_1 \neq 0$, i.e., $\alpha = kg_i \notin I_{\{f_1\}}$. So $d(I_{\{f_1\}})\geq 2$. Set $\alpha = g_2-g_4$. Since $\alpha f_1 = (g_2-g_4)f_1 = 0$, we get $\alpha = g_2-g_4 \in I_{\{f_1\}}$. Hence, $d(I_{\{f_1\}}) = 2$.

(3) Clearly, $I_{\{f_i\}} = \{\alpha \in \Bbb F_qS_4|\alpha f_i = 0\}, i = 2, 3$. Firstly, we will give the dimension of group code $I_{\{f_2\}}$. For any $\alpha = \sum\limits_{i = 1}^{24}a_ig_i\in I_{\{f_2\}}$, we have $\alpha f_2 = 0$. Then, we obtain a corresponding system of equations:

Since the rank of the coefficient matrix of the above equations is 9, we know that $\dim(I_{\{f_2\}}) = 15$. Secondly, we will compute the minimal distance of group code $I_{\{f_2\}}$. For some $\alpha = kg_i, 1\leq i \leq 24$, if $k \neq 0$, then $\alpha f_2 \neq 0$, i.e., $\alpha = kg_i \notin I_{\{f_2\}}$. Though the coefficients of $f_2$, it is not difficult to find that $\alpha f_2 \neq 0$ for $wt(\alpha)\leq3$. So $d(I_{\{f_2\}})\geq 4$. Set $\alpha = g_1+g_2+g_3+g_4$. Since $\alpha f_1 = (g_1+g_2+g_3+g_4)f_1 = 0$, we get $\alpha = g_1+g_2+g_3+g_4 \in I_{\{f_2\}}$. Hence, $d(I_{\{f_2\}}) = 4$. We can compute the parameters of group code $I_{\{f_2\}}$ by the similarly method (for $d(I_{\{f_3\}}) = 4$, take $\alpha = -g_1+g_2-g_3+g_4$).

This completes the proof. □

Theorem 3.7. Let $e_i, i = 1, 2$ and $f_j, j = 1, 2, 3$ be idempotents in $\Bbb F_qS_4$. Then

(1) $I_{\{e_1, e_2\}}$ is a [24,22,2] group code;

(2) $I_{\{e_i, f_1\}}$ is a [24,17,2] group code, for $i = 1, 2$;

(3) $I_{\{e_i, f_j\}}$ is a [24,14,8] group code, for $i = 1, 2, j = 2, 3$;

(4) $I_{\{f_1, f_j\}}$ is a [24,11,8] group code, for $j = 2, 3$;

(5) $I_{\{f_2, f_3\}}$ is a [24,6,8] group code.

Proof. (1) Clearly, $I_{\{e_1, e_2\}} = \{\alpha \in \Bbb F_qS_4|\alpha e_1 = 0\ \mbox{and}\ \alpha e_2 = 0\}$. Firstly, we will give the dimension of group code $I_{\{e_1, e_2\}}$. For any $\alpha = \sum\limits_{i = 1}^{24}a_ig_i\in I_{\{e_1, e_2\}}$, we have $\alpha e_1 = 0$ and $\alpha e_2 = 0$. From (3.4) and (3.5), we obtain the following system of equations: $\left\{ \begin{array}{ll} \sum\limits_{i = 1}^{24}a_i = 0 \\ (a_1+a_3+a_{6}+a_{8}+a_{9}+a_{11}+a_{14}+a_{17}+a_{17}+a_{19}+a_{21}+a_{23}) \\ -(a_2+a_4+a_{5}+a_{7}+a_{10}+a_{12}+a_{13}+a_{15}+a_{18}+a_{20}+a_{22}+a_{24}) = 0 \end{array} \right.$.

Then, we have $\dim(I_{\{e_1, e_2\}}) = 22$. Secondly, we will compute the minimal distance of group code $I_{\{e_1, e_2\}}$. For some $\alpha = kg_i, 1\leq i \leq 24$, if $k \neq 0$, then $\alpha e_i \neq 0$, i.e., $\alpha = kg_i \notin I_{\{e_1, e_2\}}$. So $d(I_{\{e_1, e_2\}})\geq 2$. Set $\alpha = g_1-g_3$. Since $\alpha e_i = (g_1-g_3)e_i = 0$, for $i = 1, 2$, we get $\alpha = g_1-g_3 \in I_{\{e_1, e_2\}}$. Hence, $d(I_{\{e_1, e_2\}}) = 2$.

(2) Let $I_{\{e_i, f_1\}}, i = 1, 2$ be the set of elements of the form $\alpha = \sum\limits_{i = 1}^{24}a_{i}g_i$ which the coefficients of $\alpha$ satisfy Eqs (3.6) and (3.8)–(3.13) for $i = 1$, while Eqs (3.7)–(3.13) for $i = 2$. Obviously, it does not contain any element with weight 1 of $I_{\{e_i, f_1\}}, i = 1, 2$. Also, set $\alpha = g_1-g_3$, then $\alpha \in I_{\{e_i, f_1\}}, i = 1, 2$. Hence, $d(I_{\{e_i, f_1\}}) = 2, i = 1, 2$. Moreover, the dimension of $I_{\{e_i, f_2\}}, i = 1, 2$ is 17. Therefore, $I_{\{e_i, f_1\}}, i = 1, 2$ is a [2,17,24] group code.

(3) Let $I_{\{e_i, f_2\}}, i = 1, 2$ be the set of elements of the form $\alpha = \sum\limits_{i = 1}^{24}a_{i}g_i$ which the coefficients of $\alpha$ satisfy Eqs (3.6) and (3.14)–(3.37) for $i = 1$, while Eqs (3.7) and (3.14)–(3.37) for $i = 2$. Obviously, it does not contain any element with weight $\leq 4$ of $I_{\{e_i, f_2\}}, i = 1, 2$. Also, set $\alpha = g_1+g_2+g_3+g_4-g_5-g_6-g_7-g_8$, then $\alpha \in I_{\{e_i, f_2\}}, i = 1, 2$. Hence, $d(I_{\{e_i, f_2\}}) = 8, i = 1, 2$. Moreover, the dimension of $I_{\{e_i, f_2\}}, i = 1, 2$ is 14. Therefore, $I_{\{e_i, f_2\}}, i = 1, 2$ is a [8,14,24] group code. In addition, we can obtain the parameters of $I_{\{e_i, f_3\}}, i = 1, 2$ by a similar proof.

(4) Let $I_{\{f_1, f_2\}}$ be the set of elements of the form $\alpha = \sum\limits_{i = 1}^{24}a_{i}g_i$ which the coefficients of $\alpha$ satisfy Eqs (3.8)–(3.37). Obviously, it does not contain any element with weight $\leq 4$ of $I_{\{f_1, f_2\}}$. Also, set $\alpha = g_1+g_2+g_3+g_4-g_{21}-g_{22}-g_{23}-g_{24}$, then $\alpha \in I_{\{f_1, f_2\}}$. Hence, $d(I_{\{f_1, f_2\}}) = 8$. Moreover, the dimension of $I_{\{f_1, f_2\}}$ is 11. Therefore, $I_{\{f_1, f_2\}}$ is a [24,11,8] group code. In addition, we can obtain the parameters of $I_{\{f_1, f_3\}}$ by a similar proof.

(5) Let $I_{\{f_2, f_3\}}$ be the set of elements of the form $\alpha = \sum\limits_{i = 1}^{24}a_{i}g_i$ which the coefficients of $\alpha$ satisfy the corresponding equations. Obviously, it does not contain any element with weight $\leq 4$ of $I_{\{f_2, f_3\}}$. Also, set $\alpha = g_1+g_2+g_3+g_4-g_{21}-g_{22}-g_{23}-g_{24}$, then $\alpha \in I_{\{f_2, f_3\}}$. Hence, $d(I_{\{f_2, f_3\}}) = 8$. Moreover, the dimension of $I_{\{f_2, f_3\}}$ is 6. Therefore, $I_{\{f_2, f_3\}}$ is a [6,8,24] group code.

This completes the proof. □

We summarize the following results about the group codes of $\Bbb F_qS_4$. In fact, these results can be proved by using similar techniques with rigorous derivation that we have used in the previous results in this section.

Theorem 3.8. Let $e_i, i = 1, 2$ and $f_j, j = 1, 2, 3$ be idempotents in $\Bbb F_qS_4$. Then

(1) $I_{\{e_1, e_2, f_1\}}$ is a [24,18,2] group code;

(2) $I_{\{e_1, e_2, f_j\}}$ is a [24,13,8] group code, for $j = 2, 3$;

(3) $I_{\{e_i, f_1, f_j\}}$ is a [24,10,8] group code, for $i = 1, 2, j = 2, 3$;

(4) $I_{\{e_i, f_2, f_3\}}$ is a [24,5,8] group code, for $i = 1, 2$;

(5) $I_{\{f_1, f_2, f_3\}}$ is a [24,2,8] group code, for $i = 1, 2$.

Theorem 3.9. Let $e_i, i = 1, 2$ and $f_j, j = 1, 2, 3$ be idempotents in $\Bbb F_qS_4$. Then

(1) $I_{\{e_i, f_1, f_2, f_3\}}$ is a [24,1,8] group code, for $i = 1, 2$;

(2) $I_{\{e_1, e_2, f_1, f_j\}}$ is a [24,9,8] group code, for $j = 2, 3$;

(3) $I_{\{e_1, e_2, f_2, f_3\}}$ is a [24,4,8] group code.

We can obtain the following results based on Theorems 3.6–3.9.

Remark 3.10. Let $e_i, i = 1, 2$ and $f_j, j = 1, 2, 3$ be idempotents in $\Bbb F_qS_4$.

(1) $I_{\{e_i\}}, i = 1, 2,$ and $I_{\{e_1, e_2\}}$ are MDS group codes.

(2) There are no AMDS group codes in $\Bbb F_qS_4$.

4.   Conclusions

The main contributions of this paper are the following:

$\bullet$ The unique (linear and nonlinear) idempotents of $\Bbb F_q S_3$ and $\Bbb F_q S_4$ were described (see Theorems 3.1 and 3.6).

$\bullet$ The minimum distances and dimensions of $\Bbb F_q S_3$ and $\Bbb F_q S_4$ were characterized (see Theorems 3.2–3.3 and 3.6–3.9).

$\bullet$ The MDS group codes and almost MDS group codes in $\Bbb F_q S_3$ and $\Bbb F_q S_4$ were constructed (see Remarks 3.4 and 3.10).

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors are very grateful to the reviewers and the Associate Editor for their valuable comments and suggestions to improve the quality of this paper.

The paper is supported by National Natural Science Foundation of China (Nos. 11961050, 12101277), Fundation of Nanjing Institute of Technology (No. CKJB202007), the Guangxi Natural Science Foundation(No. 2020GXNSFAA159053).

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this paper.