GALOIS LCD CODES OVER $\boldsymbol {\mathbb {F}_q+u\mathbb {F}_q+v\mathbb {F}_q+uv\mathbb {F}_q}$

Abstract

Wu and Shi [‘A note on k-Galois LCD codes over the ring $\mathbb {F}_q + u\mathbb {F}_q$ ’, Bull. Aust. Math. Soc. 104(1) (2021), 154–161] studied $ k $ -Galois LCD codes over the finite chain ring $\mathcal {R}=\mathbb {F}_q+u\mathbb {F}_q$ , where $u^2=0$ and $ q=p^e$ for some prime p and positive integer e. We extend the results to the finite nonchain ring $ \mathcal {R} =\mathbb {F}_q+u\mathbb {F}_q+v\mathbb {F}_q+uv\mathbb {F}_q$ , where $u^2=u,v^2=v $ and $ uv=vu $ . We define a correspondence between the $ l $ -Galois dual of linear codes over $ \mathcal {R} $ and the $ l $ -Galois dual of their component codes over $ \mathbb {F}_q $ . Further, we construct Euclidean LCD and $ l $ -Galois LCD codes from linear codes over $ \mathcal {R} $ . We prove that any linear code over $ \mathcal {R} $ is equivalent to a Euclidean code over $\mathbb {F}_q$ with $ q>3 $ and an $ l $ -Galois LCD code over $ \mathcal {R}$ with $0<l<e$ and $p^{e-l}+1\mid p^e-1$ . Finally, we investigate MDS codes over $ \mathcal {R}$ .

1 Introduction

An LCD code (shortened form for linear complementary dual code) is a linear code which intersects its dual trivially. LCD codes were defined and characterised by Massey [13] over finite fields. For a two-user binary adder channel, an optimal linear coding solution is obtained by LCD codes. LCD codes have applications in many areas including consumer electronics, data storage communication systems and cryptography. Yang and Massey [16] derived LCD cyclic codes. Carlet and Guilley [2] constructed several LCD codes and presented an implementation of binary LCD codes against fault injection and side channel attacks.

Carlet et al. [5] demonstrated that any linear code over $\mathbb {F}_q\ (q> 3)$ is equivalent to a Euclidean LCD code and any linear code over $\mathbb {F}_{q^2}\ (q> 2) $ is equivalent to a Hermitian LCD code. Carlet et al. [4] characterised binary LCD codes in terms of their orthogonal or symplectic basis and proved that almost all binary LCD codes are odd-like codes with odd-like duals. Fan and Zhang [7] generalised Euclidean and Hermitian inner products to the $ l $ -Galois inner product over finite fields and studied self-dual constacyclic codes for the $ l $ -Galois inner product over finite fields. Liu et al. [11] obtained $ l $ -Galois LCD codes over finite fields, where they characterised $ \lambda $ -constacyclic codes as $ l $ -Galois LCD codes. In [12], some criteria for a linear code to be an LCD code over a finite commutative ring were obtained.

A linear code $\mathcal {C}$ with parameters $ [n,k,d] $ over a finite field is said to be a maximum distance separable (MDS) code if the minimum distance $ d $ of the code $\mathcal {C}$ attains the Singleton bound, that is, $ d=n-k+1 $ . MDS codes have very good theoretical and practical properties. Jin [8] used generalised Reed–Solomon codes to create numerous classes of LCD MDS codes. Extending this work, Chen [6] proposed an alternative method to construct new LCD MDS codes from generalised Reed–Solomon codes. Carlet et al. [3] discussed the existence of Euclidean LCD MDS codes over a finite field and gave several constructions of Euclidean and Hermitian LCD MDS codes. Li et al. [10] studied linear codes over the ring $ \mathbb {Z}_4+u\mathbb {Z}_4+v\mathbb {Z}_4+uv\mathbb {Z}_4 $ for the Euclidean inner product and discussed some properties of Euclidean dual and MDS codes. Several authors investigated skew cyclic codes, constacyclic codes and quantum error correcting codes over the ring $ \mathcal {R} $ [1, 9, 17]. Prakash et al. [14] enumerated self-dual and LCD double circulant codes over a class of finite commutative nonchain rings $R_q$ and investigated the algebraic structure of 1-generator quasi-cyclic (QC) codes over $R_q$ for $q=3$ . The $ l $ -Galois LCD codes over the finite chain ring $ \mathbb {F}_q+u\mathbb {F}_q $ are studied in [15], showing that for any linear code over $ \mathbb {F}_q+u\mathbb {F}_q $ , there exist equivalent Euclidean and $ l $ -Galois LCD codes. Taking inspiration from [15], we consider $ l $ -Galois linear codes over the finite nonchain ring, $ \mathcal {R}=\mathbb {F}_q+u\mathbb {F}_q+v\mathbb {F}_q+uv\mathbb {F}_q $ and characterise $ l $ -Galois LCD codes over this ring.

Section 2 contains the basic mathematical background we require. We define an inner product which is a generalisation of Euclidean and Hermitian inner products over $\mathcal {R}$ . In Section 3, we construct $ l $ -Galois LCD codes over $ \mathcal {R} $ . We also discuss basic results on the Gray image of an l-Galois LCD code and its dual. In Section 4, we construct Euclidean and $ l $ -Galois LCD codes from linear codes over $ \mathcal {R} $ . Moreover, we demonstrate that a linear code over $ \mathcal {R} $ is equivalent to a Euclidean and an $ l $ -Galois LCD code over $ \mathcal {R} $ . In Section 5, we look at MDS codes over $ \mathcal {R}$ and give results connecting $\mathcal {C}^{\perp _l}$ and $\mathcal {C}$ whenever one of them is an MDS code. The l-Galois LCD MDS codes over $ \mathcal {R} $ seem worthy of further study.

2 Preliminaries

Throughout, $ q $ denotes a prime power, that is, $q=p^e$ for some integer $ e>0 $ , and $ \mathbb {F}_q $ the finite field of order $ q $ . Let us consider the ring $\mathcal {R}=\mathbb {F}_q+u\mathbb {F}_q+v\mathbb {F}_q+uv\mathbb {F}_q=\{a+ub+vc+uvd \mid u^2=u, v^2=v, uv=vu, a,b,c,d \in \mathbb {F}_q \}$ . It is easy to see that $\mathcal {R}$ is a commutative principal ideal ring. Since it has four maximal ideals, it is a semi-local ring and a finite nonchain ring. Let $\gamma _1=1-u-v+uv,\ \gamma _2=uv,\ \gamma _3=u-uv,\ \gamma _4=v-uv$ , so that $\sum _{i=1}^4 \gamma _i=1$ , $\gamma _i^2=\gamma _i$ and $\gamma _i\gamma _j=0$ for $i\neq j$ . By the Chinese remainder theorem, $\mathcal {R}=\gamma _1\mathcal {R}\oplus \gamma _2\mathcal {R}\oplus \gamma _3\mathcal {R}\oplus \gamma _4\mathcal {R}$ and $\gamma _i\mathcal {R}\cong \gamma _i\mathbb {F}_q $ for $ i=1,2,3,4$ . For any $ a\in \mathcal {R} $ , $ a $ can be uniquely written as $ a=\sum _{i=1}^{4}\gamma _i a_i $ , where $ a_i\in \mathbb {F}_q $ for $ i=1,2,3,4 $ . Hence, $\mathcal {R}\cong \gamma _1\mathbb {F}_q\oplus \gamma _2\mathbb {F}_q\oplus \gamma _3\mathbb {F}_q\oplus \gamma _4\mathbb {F}_q$ .

Definition 2.1. A code $ \mathcal {C} $ over $\mathcal {R}$ of length n is a nonempty subset of $\mathcal {R}^n$ . The code $\mathcal {C}$ is said to be linear if it is an $\mathcal {R}$ -submodule of $\mathcal {R}^n$ .

Definition 2.2. The Hamming weight $wt_H(x)$ of $x=(x_1,x_2,\ldots ,x_n) \in \mathbb {F}_q^n$ is the number of nonzero $x_i$ for $i\in \{1,2,\ldots ,n\}$ . For $y\in \mathbb {F}_q^n$ , the Hamming distance between x and y is the Hamming weight of the vector $x-y$ .

Definition 2.3. The Hamming distance of a code $\mathcal {C}$ , denoted by $d_H(\mathcal {C})$ , is the number $d_H(\mathcal {C})=\min \{wt_H(x) \text { | }x\neq 0\}$ .

Definition 2.4. For $ r=a_1+a_2u+a_3v+a_4uv\in \mathcal {R} $ , the Lee weight of $ r $ is $ wt_L(r)=wt_H(a_1,a_1+a_2,a_1+a_3,a_1+a_2+a_3+a_4) $ . The definition of Lee weight can be extended to $\mathcal {R}^n$ : for $s=(s_1,s_2,\ldots ,s_n) \in \mathcal {R}^n $ , the Lee weight of s is $wt_L(s)=\sum _{i=1}^{n} wt_L(s_i)$ . If $t=(t_1,t_2,\ldots ,t_n) \in \mathcal {R}^n $ , then the Lee distance between the two vectors $ s $ and $ t $ is $ d_L(s,t)=wt_L(s-t)=\sum _{i=1}^{n}wt_L(s_i-t_i).$

Definition 2.5. The Lee distance of the code $ \mathcal {C} $ , denoted by $d_L(\mathcal {C})$ , is the number $ d_L(\mathcal {C})=\min \{d_L(s-t) \text { | }s\neq t\}.$

A function $ \rho : \mathcal {R} \mapsto \mathbb {F}_q^4 $ is a Gray map if it is bijective and distance preserving. From [17], the function $\rho : \mathcal {R}\to \mathbb {F}_q^4$ defined by

$$ \begin{align*} \rho(r)=\rho(a_1+a_2u+a_3v+a_4uv)=(a_1,a_1+a_2,a_1+a_3,a_1+a_2+a_3+a_4)\end{align*} $$

is a Gray map. An equivalent Gray map for $ r=\sum _{i=1}^{4} \gamma _ir_i\in \mathcal {R}$ , where $ r_i\in \mathbb {F}_q $ for $i=1,2,3,4 $ , is

$$ \begin{align*}\rho(r)=\rho\bigg( \sum_{i=1}^{4}\gamma_i r_i\bigg) =(r_1,r_2,r_3,r_4).\end{align*} $$

We can easily extend this to a map from $\mathcal {R}^n$ to $\mathbb {F}_q^{4n}$ . By the definition of the Gray map, $ \rho $ is linear over $ \mathbb {F}_q $ and it preserves distance from $ (\mathcal {R}^n,d_L) $ to $ (\mathbb {F}_q^{4n},d_H) $ , where $d_L$ is the Lee distance and $d_H$ is the Hamming distance. The following result can be obtained directly from the definition of $\rho $ .

Proposition 2.6. For a linear code $\mathcal {C}$ of length n over the ring $\mathcal {R}$ with cardinality $q^k$ and Lee distance d, $ \rho (\mathcal {C}) $ is a $[4n,k,d]$ linear code over $\mathbb {F}_q$ .

Define a Frobenius operator $F:\mathcal {R}\to \mathcal {R}$ over $\mathcal {R}$ by

$$ \begin{align*} F(a_1+a_2u+a_3v+a_4uv)=a_1^p+ua_2^p+va_3^p+uva_4^p.\end{align*} $$

Equivalently, $F(r)=\gamma _1r_1^p+\gamma _2r_2^p+\gamma _3r_3^p+\gamma _4r_4^p$ for $r=\sum _{i=1}^{4}\gamma _i r_i\in \mathcal {R}$ .

For $ s=(s_1,s_2,\ldots ,s_n) $ and $ t=(t_1,t_2,\ldots ,t_n) \in \mathcal {R}^n$ and $ 0\leq l\leq e-1 $ , define the l-Galois inner product,

$$ \begin{align*} [s,t]_l=\sum_{i=1}^{n}s_iF^{l}(t_i) .\end{align*} $$

Remark 2.7. This inner product is a generalisation of the Euclidean and Hermitian inner products for $ l=0 $ and $ l={e}/{2} $ (when $ e $ is even), respectively.

From now on, we write $ [s,t], [s,t]_H $ and $ [s,t]_l $ for the Euclidean, Hermitian and $ l $ -Galois inner product over $ \mathcal {R} $ and $ \langle s,t\rangle , \langle s,t\rangle _H $ and $ \langle s,t\rangle _l $ for the Euclidean, Hermitian and $ l $ -Galois inner product over $ \mathbb {F}_q $ , respectively. The $ l$ -Galois dual code $\mathcal {C}^{\perp _l}$ of $\mathcal {C}$ over $\mathcal {R}$ is defined by

$$ \begin{align*}\mathcal{C}^{\perp_l}=\{s\in \mathcal{R}^n \mid [t,s]_l=0\mbox{ for all } t\in \mathcal{C}\}.\end{align*} $$

Clearly, $\mathcal {C}^{\perp _l}$ is a linear code over $\mathcal {R}$ . A linear code over $ \mathcal {R} $ is said to be $ l $ -Galois LCD if $ \mathcal {C}\cap \mathcal {C}^{\perp _l}=\{0\} $ . It is well known that for a Frobenius ring $ \mathcal {R} $ and a linear code $\mathcal {C}$ over the ring $ \mathcal {R} $ of length $ n $ , the product of the cardinalities of $\mathcal {C}$ and $\mathcal {C}^{\perp _l}$ is equal to the cardinality of $ \mathcal {R}^n $ , that is, $ |\mathcal {C}||\mathcal {C}^{\perp _l}|=|\mathcal {R}^n| .$

Remark 2.8. For $ l=0 $ and $ l = e/2 $ (when $ e $ is even), this construction gives the Euclidean and the Hermitian dual code, respectively.

3 $ { l } $ -Galois linear codes over $ \mathbf {\mathcal {R}} $

In this section, we derive a necessary and sufficient condition for $\mathcal {C}$ to be an $ l$ -Galois LCD code over $ \mathcal {R} $ with respect to its component codes. Also, we give a relationship between an l-Galois LCD code and its Gray image.

A linear code $\mathcal {C}$ over $ \mathcal {R} $ can be decomposed into four component codes over the finite field $ \mathbb {F}_q $ as follows:

$$ \begin{align*} \mathcal{C}_1=\{x\in \mathbb{F}_q^n \mid \gamma_1x+\gamma_2y+\gamma_3z+\gamma_4w\in \mathcal{C} \text{ for some } y,z,w\in \mathbb{F}_q^n\},\\ \mathcal{C}_2=\{y\in \mathbb{F}_q^n \mid \gamma_1x+\gamma_2y+\gamma_3z+\gamma_4w\in \mathcal{C} \text{ for some } x,z,w \in \mathbb{F}_q^n\},\\ \mathcal{C}_3=\{z\in \mathbb{F}_q^n \mid \gamma_1x+\gamma_2y+\gamma_3z+\gamma_4w\in \mathcal{C} \text{ for some } x,y,w \in \mathbb{F}_q^n\},\\ \mathcal{C}_4=\{w\in \mathbb{F}_q^n \mid \gamma_1x+\gamma_2y+\gamma_3z+\gamma_4w\in \mathcal{C} \text{ for some } x,y,z \in \mathbb{F}_q^n\}. \end{align*} $$

The $\mathcal {C}_i$ are linear codes over $ \mathbb {F}_q $ for $ 1\leq i\leq 4$ and $\mathcal {C}=\gamma _1 \mathcal {C}_1\oplus \gamma _2 \mathcal {C}_2\oplus \gamma _3\mathcal {C}_3\oplus \gamma _4 \mathcal {C}_4$ . We say $ \mathcal {C}_1,\mathcal {C}_2,\mathcal {C}_3 $ and $ \mathcal {C}_4 $ are component codes of the linear code $ \mathcal {C} $ . The cardinality of a linear code $\mathcal {C}$ is the product of the cardinalities of its component codes, that is,

$$ \begin{align*}|\mathcal{C}|=|\mathcal{C}_1||\mathcal{C}_2||\mathcal{C}_3||\mathcal{C}_4|.\end{align*} $$

The Lee distance of a linear code $\mathcal {C}$ is the minimum of the Hamming distances of its component codes,

$$ \begin{align*}d_L(\mathcal{C})=\min_{1\leq i\leq 4}\{d_H(\mathcal{C}_i)\}.\end{align*} $$

Let $\mathcal {C}^{p^l}=\{(F^l(c_1),F^l(c_2),\ldots ,F^l(c_n)) \mid (c_1,c_2,\ldots ,c_n)\in \mathcal {C}\}$ and $F^l(G)=(F^l(g_{ij}))$ for a matrix $ G=(g_{ij}) $ over $ \mathcal {R} $ . For $ 1\leq i \leq 4 $ , let $G_i$ be the generator matrix for $\mathcal {C}_i$ . Then the generator matrices for $\mathcal {C}$ and $\rho (\mathcal {C})$ are

$$ \begin{align*}G=\begin{bmatrix} \gamma_1G_1\\ \gamma_2G_2\\ \gamma_3G_3\\ \gamma_4G_4 \end{bmatrix} \quad\mbox{and}\quad \rho(G)=\begin{bmatrix} \,\rho(\gamma_1G_1)\\ \,\rho(\gamma_2G_2)\\ \,\rho(\gamma_3G_3)\\ \,\rho(\gamma_4G_4) \end{bmatrix},\end{align*} $$

where $\rho (\gamma _iG_i)$ is a matrix over $\mathbb {F}_q$ for $ 1\leq i\leq 4. $ Since $ \gamma _i\gamma _j=0 $ for $ i\neq j $ and $ \gamma _i^2=\gamma _i $ for $ i=1,2,3,4 $ ,

$$ \begin{align*} G(F^{e-l}(G))^T=&\begin{bmatrix} \gamma_1 G_1F^{e-l}(G_1)^T & 0 &0 &0\\ 0&\gamma_2 G_2F^{e-l}(G_2)^T &0&0\\ 0&0&\gamma_3 G_3F^{e-l}(G_3)^T&0\\ 0&0&0&\gamma_4 G_4F^{e-l}(G_4)^T \end{bmatrix}\!. \end{align*} $$

We call $\mathcal {C}$ an $ [n,k,d] $ code over $ \mathcal {R} $ if $\mathcal {C}$ is a code of length $ n $ , $ |\mathcal {C}|=q^k $ and $ d $ is the Lee distance. If the $\mathcal {C}_i $ are the component codes of $\mathcal {C}$ , with parameters $ [n,k_i,d_i] $ for $ i=1,2,3,4 $ , then $ k=\sum _{i=1}^4k_i $ and $ d= \min _{1\leq i\leq 4}\{d_i\} $ .

In the following lemma, we observe that the Euclidean dual of $\mathcal {C}^{p^{(e-l)}}$ is equal to the l-Galois dual code of $\mathcal {C}$ .

Lemma 3.1. If $\mathcal {C}$ is an $ [n,k,d] $ linear code over $ \mathcal {R} $ , then $ \mathcal {C}^{p^{(e-l)}} $ is an $[n,k,d]$ linear code over $ \mathcal {R} $ and $ \mathcal {C}^{\perp _l}=(\mathcal {C}^{p^{(e-l)}}) ^\perp .$ Moreover, if $\mathcal {C}$ has generator matrix G, then $ F^{e-l}(G)$ is a generator matrix of $\mathcal {C}^{p^{(e-l)}}$ .

The next theorem gives the decomposition of the l-Galois dual code into its component codes. Consequently, we obtain a relation between l-Galois LCD codes over $ \mathcal {R} $ and l-Galois LCD component codes over $ \mathbb {F}_q$ .

Theorem 3.2. For a linear code $ \mathcal {C}= \bigoplus _{i=1}^4\gamma _i \mathcal {C}_i $ over $ \mathcal {R} $ :

1. (1) $ \mathcal {C}^{\perp _l}= \bigoplus _{i=1}^4\gamma _i \mathcal {C}_i^{\perp _l} $ ;

2. (2) $\mathcal {C}$ is an $ l $ -Galois LCD code over $ \mathcal {R} $ if and only if all its component codes $ \mathcal {C}_i $ are $ l $ -Galois LCD codes over $ \mathbb {F}_q $ for $ 1\leq i\leq 4$ ;

3. (3) $\mathcal {C}$ is an $ l $ -Galois self-orthogonal linear code over $\mathcal {R}$ if and only if all its component codes $ \mathcal {C}_i $ are $ l $ -Galois self-orthogonal codes over $ \mathbb {F}_q$ and $\mathcal {C}$ is a self-dual code if and only if all its component codes $ \mathcal {C}_i $ are self-dual codes over $ \mathbb {F}_q $ for $ 1\leq i\leq 4. $

Proof. (1) If $ x=\gamma _1x_1+\gamma _2x_2+\gamma _3x_3+\gamma _4x_4\in \mathcal {C}^{\perp _l}$ , then $ [\,y,x]_l=0 $ for any $ y=\gamma _1y_1+\gamma _2y_2+\gamma _3y_3+\gamma _4y_4\in \mathcal {C}.$ Since $ \gamma _i^2=\gamma _i$ and $ \gamma _i\gamma _j=0 $ for $ i\neq j,\ [\,y,x]_l=\gamma _1\langle y_1,x_1\rangle _l+\gamma _2\langle y_2,x_2\rangle _l+\gamma _3\langle y_3,x_3\rangle _l+\gamma _4\langle y_4,x_4\rangle _l $ . Thus, $ \langle y_i,x_i\rangle _l=0 $ for all $ y_i\in \mathcal {C}_i $ and $ i=1,2,3,4 $ , that is, $ x_i\in \mathcal {C}_i^{\perp _l} $ for $ i=1,2,3,4 $ . Therefore, $ x\in \gamma _1 \mathcal {C}_1^{\perp _l}\oplus \gamma _2 \mathcal {C}_2^{\perp _l}\oplus \gamma _3 \mathcal {C}_3^{\perp _l}\oplus \gamma _4 \mathcal {C}_4^{\perp _l} $ .

Conversely, let $ w=\gamma _1w_1+\gamma _2w_2+\gamma _3w_3+\gamma _4w_4\in \gamma _1 \mathcal {C}_1^{\perp _l}\oplus \gamma _2 \mathcal {C}_2^{\perp _l}\oplus \gamma _3 \mathcal {C}_3^{\perp _l}\oplus \gamma _4 \mathcal {C}_4^{\perp _l} $ , where $ w_i\in \mathcal {C}_i^{\perp _l} $ . For any $ y=\gamma _1y_1+\gamma _2y_2+\gamma _3y_3+\gamma _4y_4\in \mathcal {C}$ , where $ y_i\in \mathcal {C}_i$ , $[\,y,w]_l=\gamma _1\langle y_1,w_1\rangle _l+\gamma _2\langle y_2,w_2\rangle _l+\gamma _3\langle y_3,w_3\rangle _l+\gamma _4\langle y_4,w_4\rangle _l=0 $ , which implies that $ w\in \mathcal {C}^{\perp _l}.$ Hence, $ \mathcal {C}^{\perp _l}= \gamma _1 \mathcal {C}_1^{\perp _l}\oplus \gamma _2 \mathcal {C}_2^{\perp _l}\oplus \gamma _3 \mathcal {C}_3^{\perp _l}\oplus \gamma _4 \mathcal {C}_4^{\perp _l} $ .

(2) Suppose that $\mathcal {C}$ is an $ l $ -Galois LCD code over $ \mathcal {R} $ , that is, $ \mathcal {C}\cap \mathcal {C}^{\perp _l}=\{0\}$ . Let $ x_i\in \mathcal {C}_i\cap \mathcal {C}_i^{\perp _l} $ for some $ i=1,2,3,4 $ , that is, $ \langle y_i,x_i\rangle _l=0 $ for all $ y_i\in \mathcal {C}_i $ . Now take $ x=\gamma _ix_i \in \mathcal {C}$ . Then for any $ y=\gamma _1y_1+\gamma _2y_2+\gamma _3y_3+\gamma _4y_4\in \mathcal {C}$ , where $y_j\in \mathcal {C}_j $ for $j=1,2,3,4 $ , $ [\,y,x]_l=[\gamma _1y_1+\gamma _2y_2+\gamma _3y_3+\gamma _4y_4,\gamma _ix_i]_l=\gamma _i\langle y_i,x_i \rangle _l=0$ , since $ \gamma _i^2=\gamma _i$ and $ \gamma _i\gamma _j=0 $ for $i\neq j .$ This implies that $ x\in \mathcal {C}\cap \mathcal {C}^{\perp _l}=\{0\} $ , that is, $ x=0 $ , consequently, $ x_i=0 .$ Hence, $ \mathcal {C}_i $ is an $ l $ -Galois LCD code over $ \mathbb {F}_q $ .

Conversely, suppose the $ \mathcal {C}_i $ are $ l $ -Galois LCD codes over $ \mathbb {F}_q $ for $ i=1,2,3,4 $ . Let $ x=\gamma _1x_1+\gamma _2x_2+\gamma _2x_3+ \gamma _4x_4\in \mathcal {C}\cap \mathcal {C}^{\perp _l}$ . Then $ x_i \in \mathcal {C}_i\cap \mathcal {C}_i^{\perp _l} $ and $\mathcal {C}_i\cap \mathcal {C}_i^{\perp _l}=\{0\} $ , which implies that $ x=0. $ Thus, $\mathcal {C}$ is an $ l $ -Galois LCD code.

The proof of part (3) follows easily from part (1), so we omit the proof.

Remark 3.3. Parts (1) and (3) in Theorem 3.2 have been proved for the Euclidean dual over the ring $ \mathbb {Z}_4 +u\mathbb {Z}_4+v\mathbb {Z}_4+uv\mathbb {Z}_4$ in [10].

The next corollary gives a necessary and sufficient condition for an l-Galois LCD code over $\mathcal {R}$ in terms of generator matrices.

Corollary 3.4. Let $ \mathcal {C}=\bigoplus _{i=1}^4\gamma _i \mathcal {C}_i$ with generator matrix

$$ \begin{align*}G=\begin{bmatrix} \gamma_1G_1\\ \gamma_2G_2\\ \gamma_3G_3\\ \gamma_4G_4 \end{bmatrix},\end{align*} $$

where $ G_i $ is a generator matrix for $ \mathcal {C}_i $ over $ \mathbb {F}_q $ . Then, $\mathcal {C}$ is an $ l $ -Galois LCD code over $ \mathcal {R} $ if and only if the matrix $ G_i(F^{e-l}(G_i))^T $ is nonsingular for $ i=1,2,3,4 $ over $ \mathbb {F}_q $ .

Proof. By Theorem 3.2, $\mathcal {C}$ is an $ l $ -Galois LCD code if and only if the $ \mathcal {C}_i $ are $ l $ -Galois LCD codes over $ \mathbb {F}_q$ . From [11, Theorem 2.4], $ \mathcal {C}_i $ is an $ l $ -Galois LCD code if and only if $ G_i(F^{e-l}(G_i))^T $ is nonsingular over $ \mathbb {F}_q $ .

Now, by using the definition of $\rho $ , we derive some useful properties of the Gray image of $ l $ -Galois dual codes over $ \mathcal {\mathcal {R}} $ .

Lemma 3.5. If $ \mathcal {C} $ is an $ [n,k] $ linear code over $ \mathcal {R} $ , then $\rho (\mathcal {C}^{\perp _l})={\rho (\mathcal {C})}^{\perp _l}.$

Proof. If $\rho (x)\in \rho (\mathcal {C}^{\perp _l}) $ , where $x\in \mathcal {C}^{\perp _l}$ , then $[z,x]_l=0$ for all $z \in \mathcal {C}$ . Let $z=\sum _{i=1}^{4}\gamma _iz_i$ and $x=\sum _{i=1}^{4}\gamma _ix_i$ , where $z_i\in \mathcal {C}_i$ and $x_i\in \mathcal {C}_i^{\perp _l}$ , so that $[z,x]_l=\gamma_1\left\langle z_1,x_1\right\rangle_l+\gamma_2\left\langle z_2,x_2\right\rangle_l+\gamma_3\left\langle z_3,x_3\right\rangle_l+\gamma_4\left\langle z_4,x_4\right\rangle_l =0$ . Hence, $\left\langle z_i,x_i\right\rangle_l=0$ for $i=1,2,3,4$ . Now, $\left\langle \rho(z),\rho(x)\right\rangle _l=\sum_{i=1}^{4}z_i\cdot x_i^{p^l}=\sum_{i=1}^{4}\left\langle z_i,x_i\right\rangle_l=0$ for all $\rho (z)\in \rho (\mathcal {C})$ . Thus, $\rho (x)\in {\rho (\mathcal {C})}^{\perp _l}$ . Therefore, $\rho (\mathcal {C}^{\perp _l})\subseteq {\rho (\mathcal {C})}^{\perp _l}$ .

Conversely, the cardinality of $\rho (\mathcal {C}^{\perp _l})$ is equal to $\mathcal {C}^{\perp _l}$ , that is, $|\rho (\mathcal {C}^{\perp _l})|={q^{4n}}/{|\mathcal {C}|}$ . Moreover, $|\rho (\mathcal {C})^{\perp _l}|={q^{4n}}/{|\rho (\mathcal {C})|}={q^{4n}}/{|\mathcal {C}|}$ . Hence, $ \rho (\mathcal {C}^{\perp _l})={\rho (\mathcal {C})}^{\perp _l} $ .

Lemma 3.6. If $ \mathcal {C} $ is a linear code over $ \mathcal {R} $ , then $\rho (\mathcal {C}\cap \mathcal {C}^{\perp _l})=\rho (\mathcal {C})\cap \rho (\mathcal {C}^{\perp _l})$ .

Proof. If $ \rho (x)\in \rho (\mathcal {C}\cap \mathcal {C}^{\perp _l})$ for some $x\in \mathcal {C}\cap \mathcal {C}^{\perp _l}$ , then $\rho (x)\in \rho (\mathcal {C})\cap \rho (\mathcal {C}^{\perp _l})$ . Hence, $ \rho (\mathcal {C}\cap \mathcal {C}^{\perp _l})\subseteq \rho (\mathcal {C})\cap \rho (\mathcal {C}^{\perp _l})$ . Conversely, if $ y \in \rho (\mathcal {C})\cap \rho (\mathcal {C}^{\perp _l}) $ , then $ y\in \rho (\mathcal {C}) \text {and} \ y\in \rho (\mathcal {C}^{\perp _l}) $ . Since $\rho $ is bijective, there is a unique $x\in \mathcal {C}\cap \mathcal {C}^{\perp _l}$ such that $\rho (x)=y$ . Hence, $ \rho (\mathcal {C})\cap \rho (\mathcal {C}^{\perp _l})\subseteq \rho (\mathcal {C}\cap \mathcal {C}^{\perp _l}) $ . Therefore, $\rho (\mathcal {C}\cap \mathcal {C}^{\perp _l})=\rho (\mathcal {C})\cap \rho (\mathcal {C}^{\perp _l})$ .

Theorem 3.7. A linear code $\mathcal {C}$ is an $ l $ -Galois LCD code over $ \mathcal {R} $ if and only if $ \rho (\mathcal {C})$ is an $ l $ -Galois LCD code over $ \mathbb {F}_q $ .

Proof. Suppose $\mathcal {C}$ is an $ l $ -Galois LCD code over $ \mathcal {R} $ , that is, $\mathcal {C}\cap \mathcal {C}^{\perp _l}=\{0\}$ . From Lemma 3.6, $\rho (\mathcal {C})\cap \rho (\mathcal {C})^{\perp _l}=\{0\}$ . Conversely, if $ \rho (\mathcal {C})$ is an $ l $ -Galois LCD code, then

$$ \begin{align*}\{0\}=\rho(\mathcal{C})\cap\rho(\mathcal{C})^{\perp_l}=\rho(\mathcal{C})\cap\rho(\mathcal{C}^{\perp_l})=\rho(\mathcal{C}\cap \mathcal{C}^{\perp_l}),\end{align*} $$

so that $\mathcal {C}\cap \mathcal {C}^{\perp _l}=\{0\}$ . Therefore, $\mathcal {C}$ is an $ l $ -Galois LCD code over $ \mathcal {R}. $

4 Construction of a Galois LCD code equivalent to a linear code

We give a construction of Euclidean and $ l $ -Galois LCD codes over $ \mathcal {R} $ with the help of their component codes over $\mathbb {F}_q$ . We show that for every linear code $\mathcal {C}$ , there exists a Euclidean LCD code and an $ l $ -Galois LCD code which are equivalent to $ \mathcal {C} $ .

Let m and w be integers with $ m\geq 1 $ and $ 0\leq w\leq m $ and let b be an element in $ \mathbb {F}_q^m $ with Hamming weight $ w $ . The support of b is the set $ S= \{i_1,i_2,\ldots , i_w\} $ of indices at which the components of b are nonzero. Denote the $ m\times m $ diagonal matrix whose entries are $ b_1,b_2,\ldots , b_m $ by $ \mathrm {diag}_m[b] $ . For an $ m\times m $ square matrix P over $ \mathbb {F}_q$ , let $ P_S $ denote the submatrix of $ P $ obtained by deleting the $ i_1,i_2,\ldots ,i_w $ th columns and rows of $ P $ . We write $ P_S=I $ if $ S=\{1,2,\ldots ,m\} $ and $ P_{\emptyset } =P$ .

Lemma 4.1 [5].

Let P be an $ m\times m $ matrix over $ \mathbb {F}_q $ and t an integer with $ {0\leq t\leq m-1.}$ Assume that $\det (P_J)=0$ for any $ J\subseteq \{1,2,\ldots ,m\} $ with $ 0\leq | J|\leq t$ . Then for every element $b \in \mathbb {F}_q^m$ with support S and Hamming weight $ w $ such that $ 1\leq w\leq t+1$ ,

$$ \begin{align*}\det \left( P+\mathrm{diag}_m[b]\right) =\bigg(\prod_{i\in S}b_i\bigg) \det(P_S).\end{align*} $$

Fix $ \alpha =(\alpha _1,\alpha _2,\ldots , \alpha _n)\in \mathcal {R}^n$ , where $ \alpha _j=\sum _{i=1}^{4}\gamma _i\alpha _{ji}, \ \alpha _{ji}\in \mathbb {F}_q$ for $ j=1,2,\ldots ,n $ . Define

$$ \begin{align*} \mathcal{C}^{\alpha}=\{\alpha\cdot c \mid c\in \mathcal{C}\}=\{(\alpha_1c_1,\alpha_2c_2,\ldots, \alpha_nc_n) \mid (c_1,c_2,\ldots,c_n)\in \mathcal{C}\} .\end{align*} $$

Clearly, $ \mathcal {C}^{\alpha } $ is a linear code over $ \mathcal {R} $ . Let

$$ \begin{align*} G=\begin{bmatrix} \gamma_1G_1\\ \gamma_2G_2\\ \gamma_3G_3\\ \gamma_4G_4 \end{bmatrix} \quad\mbox{and}\quad G^{\alpha}=\begin{bmatrix} \gamma_1G_1^{\alpha_1'}\\ \gamma_2G_2^{\alpha_2'}\\ \gamma_3G_3^{\alpha_3'}\\ \gamma_4G_4^{\alpha_4'} \end{bmatrix}\end{align*} $$

be generator matrices for $\mathcal {C}$ and $ \mathcal {C}^{\alpha } $ , where $G^\alpha $ is obtained by multiplying the $ j $ th column of $ G $ by $ \alpha _j $ and $ G_i^{\alpha _i'} $ is the matrix obtained by multiplying the $ j $ th column of $ G_i $ by $ \alpha _{ji} $ and $\alpha _i'=(\alpha _{1i},\alpha _{2i},\ldots ,\alpha _{ni})\in \mathbb {F}_q^n $ for $ i=1,2,3,4 $ .

Remark 4.2. Note that, if $\alpha =(\alpha _1,\alpha _2,\ldots , \alpha _n)\in \mathcal {R}^n$ and $ \alpha _j\neq 0 $ for $ 1\leq j\leq n $ , then $\mathcal {C}$ and $ \mathcal {C}^{\alpha } $ are equivalent codes over $ \mathcal {R} $ .

The following theorem gives a construction of Euclidean LCD codes over $ \mathcal {R} $ from linear codes over $ \mathcal {R} $ . We denote the parameters of the component codes $ C_i $ by $ [n,k_i,d_i] $ for $ i=1,2,3,4 $ .

Theorem 4.3. All notation is as above. Let $ \mathcal {C}=\bigoplus _{i=1}^{4}\gamma _i\mathcal {C}_i$ be an $ [n,k,d] $ linear code over $ \mathcal {R} $ , where the component codes $ \mathcal {C}_i $ over $ \mathbb {F}_q $ have generator matrices $ G_i=[I_{k_i}:M_i] $ . Let $ P_i=G_iG_i^T$ and $ t_i\leq k_i-1 $ be nonnegative integers such that $ \det ((P_i)_{S_i})=0 $ for any $ S_i \subseteq \{1,2,\ldots , k_i\} $ with $ 0\leq |S_i|\leq t_i $ and assume there exist $ R_i\subseteq \{1,2,\ldots , k_i\} $ with cardinality $ t_i+1 $ such that $ \det ((P_i)_{R_i}) \neq 0$ . If $\alpha \in \mathcal {R}^n$ is such that $ \alpha _{ji}\in \mathbb {F}_q\setminus \{+1,-1\} $ if $ j\in R_i $ and $ \alpha _{ji}\in \{+1,-1\} $ if $ j\in \{1,2,\ldots , n\}\setminus R_i$ for $\ i=1,2,3,4$ , then $ \mathcal {C}^{\alpha } $ is a Euclidean LCD code over $ \mathcal {R} $ .

Proof. Let $\alpha =(\alpha _1,\alpha _2,\ldots , \alpha _n)\in \mathcal {R}^n$ and $ c=(c_1,c_2,\ldots ,c_n)\in \mathcal {C} $ . For $\alpha _j,c_j\in \mathcal {R}$ , we write $ \alpha _j=\sum _{i=1}^{4}\gamma _i\alpha _{ji}, \ \alpha _{ji}\in \mathbb {F}_q$ , and $ c_j=\sum _{i=1}^{4} \gamma _i c_{ji} ,\ c_{ji}\in \mathcal {C}_i$ . We note that $ \alpha _jc_j=(\sum _{i=1}^{4}\gamma _i\alpha _{ji})(\sum _{i=1}^{4} \gamma _i c_{ji})=\sum _{i=1}^{4}\gamma _i\alpha _{ji}c_{ji} $ , since $ \gamma _i\gamma _m=0 $ for $ i\neq m $ and $ \gamma _i^2=\gamma _i $ for $ i,m=1,2,3,4 $ and $j=1,2,\ldots ,n.$ Now,

$$ \begin{align*} \mathcal{C}^{\alpha} & =\{(\alpha_1c_1,\alpha_2c_2,\ldots, \alpha_nc_n) \mid (c_1,c_2,\ldots,c_n)\in \mathcal{C}\}\\ & =\bigg\lbrace \bigg(\sum_{i=1}^{4}\gamma_i \alpha_{1i}c_{1i},\sum_{i=1}^{4}\gamma_i \alpha_{2i}c_{2i},\ldots ,\sum_{i=1}^{4}\gamma_i \alpha_{ni}c_{ni}\bigg)\mid (c_1,c_2,\ldots,c_n)\in \mathcal{C} \bigg\rbrace \\ & =\bigg\lbrace \sum_{i=1}^{4}\gamma_i(\alpha_{1i}c_{1i},\alpha_{2i}c_{2i},\alpha_{3i}c_{3i},\ldots, \alpha_{ni}c_{ni})\ \mid (c_1,c_2,\ldots,c_n)\in \mathcal{C} \bigg\rbrace = \bigoplus_{i=1}^{4}\gamma_i \mathcal{C}_i^{\alpha_i'}. \end{align*} $$

Here, $\mathcal {C}_i^{\alpha _i'}=\{(\alpha _{1i}c_{1i},\alpha _{2i}c_{2i},\alpha _{3i}c_{3i},\ldots , \alpha _{ni}c_{ni}) \mid (c_{1i},c_{2i},\ldots ,c_{ni})\in \mathcal {C}_i\}$ and $\alpha _i'=(\alpha _{1i},\alpha _{2i}, \ldots ,\alpha _{ni})$ . Clearly, the $\mathcal {C}_i^{\alpha _i'}$ are linear codes over $\mathbb {F}_q$ with generator matrices $ G_i^{\alpha _i'} $ for $i=1,2,3,4$ . Also, $ \alpha =\sum _{i=1}^{4}\gamma _i \alpha _i' $ . From [5, Theorem 5.1], each $ \mathcal {C}_i^{\alpha _i'} $ is a Euclidean LCD code over $ \mathbb {F}_q $ and by Theorem 3.2 (with $ l=0 $ ), $ \mathcal {C}^{\alpha } $ is a Euclidean LCD code.

Next, we use the technique described in [5] to establish the existence of $ \alpha $ for which $ \mathcal {C}^{\alpha } $ is a Euclidean LCD code for a given linear code $\mathcal {C}$ over $ \mathcal {R} $ .

Corollary 4.4. Let $ \mathbb {F}_q\ (q>3) $ be a finite field and $\mathcal {C}$ be an $ [n,k,d] $ linear code over $\mathcal {R}$ . Then $ \mathcal {C}^{\alpha } $ is an $ [n,k,d] $ Euclidean LCD code over $ \mathcal {R} $ for some $ \alpha =(\alpha _1,\alpha _2,\ldots , \alpha _n)$ in $ \mathcal {R}^n $ with $ \alpha _j\neq 0 $ for $ 1\leq j\leq n $ .

Proof. Let $ \mathcal {C}=\bigoplus _{i=1}^4\gamma _i\mathcal {C}_i $ be a linear code over $ \mathcal {R} $ . If $\mathcal {C}$ is a Euclidean LCD code, then we can take $ \alpha =(\alpha _1,\alpha _2,\ldots , \alpha _n) \in \mathcal {R}^n$ such that $ \alpha _j= \gamma _1+\gamma _2 +\gamma _3+\gamma _4$ for ${ 1\leq j\leq n.} $ Then, $ \mathcal {C}^{\alpha }=\mathcal {C} $ , a Euclidean LCD code over $ \mathcal {R} $ .

If $\mathcal {C}$ is not a Euclidean LCD code, then by Theorem 3.2, $\mathcal {C}_i $ is not a Euclidean LCD code for some $ i=1,2,3,4$ . If $ G_i $ is the generator matrix for $ \mathcal {C}_i $ , then $ \det (G_iG_i^T)=0 $ . Set $ P_i =G_iG_i^T$ . There exists an integer $ t_i\geq 0 $ and $ R_i \subseteq \{1,2,\ldots , k_i \} $ with cardinality $ | R_i|=t_i+1 $ such that $ \det ((P_i)_{R_i})\neq 0 $ and $ \det ((P_i)_{S_i}) =0$ for any $ S_i \subseteq \{1,2,\ldots ,k_i\} $ with $ 0\leq |S_i|\leq t_i $ . Also, $ \mathbb {F}_q^*\setminus \{-1,1\}\neq \emptyset $ since $ q>3 $ . Choose $ \alpha _i'=(\alpha _{1i},\alpha _{2i},\ldots ,\alpha _{ni}) \in \mathbb {F}_q^n $ such that $ \alpha _{ji} \in \mathbb {F}_q^*\setminus \{-1,1\} $ if $ j\in R_i $ and $ \alpha _{ji}=1 $ if $ j\in \{1,2,\ldots , k_i\}\setminus R_i$ . By [5, Theorem 5.1], $ \mathcal {C}_i^{\alpha _i'} $ is a Euclidean LCD code over $ \mathbb {F}_q $ . Take $ \alpha =\sum _{m=1}^4 \gamma _m \alpha _m' \in \mathcal {R}^n $ , where $ \alpha _m'=\alpha _i' $ for $ m=i $ and $ \alpha _m'=(1,1,\ldots ,1) $ for $ m\neq i $ . Then by Theorem 4.3 $, \mathcal {C}^{\alpha }=\bigoplus _{m=1}^4 \gamma _m\mathcal {C}_m^{\alpha _m'} $ is an $ [n,k,d ] $ Euclidean LCD code over $ \mathcal {R} $ .

Next, we construct an $ l $ -Galois LCD code from a given linear code over a finite field. Then similarly, we provide the construction over $ \mathcal {R}$ .

Theorem 4.5. Let $\mathbb {F}_q \ (q=p^e)$ be a finite field. For $0<l<e$ and $p^{e-l}+1\mid p^e-1$ , set $\beta ={(p^e-1)}/{(p^{e-l}+1)}$ . Let $G=[I_k:M]$ be a generator matrix for a linear code $\mathcal {C}$ over $ \mathbb {F}_q $ with parameters $[n,k,d]$ and denote the matrix $G{( F^{e-l}(G)) }^T$ by $ P $ . Let t with $ 0\leq t\leq k-1$ be an integer such that $\det (P_I)=0$ for any $I\subseteq \{1,2,\ldots ,k\}$ with $0\leq | I|\leq ~t$ , and assume there exist $J\subseteq \{1,2,\ldots ,k\} $ with cardinality $t+1$ such that $ \det (P_J)\neq 0 $ . Suppose $ a \in \mathbb {F}_q^n$ such that $ a_j\in \mathbb {F}_q \setminus (\mathbb {F}_q^*)^{\,\beta }$ for $ j\in J $ and $ a_j\in (\mathbb {F}_q^*)^{\,\beta }$ for $ j\in \{1,2,\ldots ,n\}\setminus J $ . Then, $ \mathcal {C}^{a} $ is an $ l $ -Galois LCD code over $ \mathbb {F}_q $ .

Proof. A generator matrix $ G^a $ for $ \mathcal {C}^a $ is obtained by multiplying the $ j $ th column of the matrix $ G=[I_k:M=(m_{is})] $ by $ a_j $ for $ 1\leq j\leq n $ . The $(ij)$ th entry of $ G^{a}(F^{e-l}(G^{a}))^T$ is $a_i^{p^{e-l}+1}+\sum _{s=1}^{n-k}a_{k+s}^{p^{e-l}+1}m_{is}^{p^{e-l}+1}$ if $i=j$ and $\sum _{s=1}^{n-k}a_{k+s}^{p^{e-l}+1}m_{is}m_{js}^{p^{e-l}}$ if $i\neq j$ . Since $a_{k+s}\notin J$ , $a_{k+s}^{p^{e-l}+1}=1$ for $1\leq s\leq n-k$ . The $(ij)$ th entry of $ G^{a}(F^{e-l}(G^{a}))^T$ is $a_i^{p^{e-l}+1}+\sum _{s=1}^{n-k}m_{is}^{p^{e-l}+1}$ if $i=j$ and $\sum _{s=1}^{n-k}m_{is}m_{js}^{p^{e-l}}$ if $i\neq j$ . The $(ij)$ th entry of $ G(F^{e-l}(G))^T$ is $1+\sum _{s=1}^{n-k}m_{is}^{p^{e-l}+1}$ if $i=j$ and $\sum _{s=1}^{n-k}m_{is}m_{js}^{p^{e-l}}$ if $i\neq j$ . Hence, $ G^{a}(F^{e-l}(G^{a}))^T=G{( F^{e-l}(G)) }^T+ \text {diag}_k[b] $ , where $ b=(a_1^{p^{e-l}+1}-1,a_2^{p^{e-l}+1}-1,\ldots , a_k^{p^{e-l}+1}-1) $ . Note that the support of $ b $ is the set $ J $ . By Lemma 4.1, $ \det (G^{a}(F^{e-l}(G^{a}))^T)=\det ( P+ \mathrm {diag}_k[b]) =(\prod _{j\in J}b_j ) \det (P_J)\neq 0 $ . Hence, $ \mathcal {C}^{a} $ is an $ l $ -Galois LCD code over $ \mathbb {F}_q $ .

Theorem 4.6. All notation is as above. For $0<l<e$ and $p^{e-l}+1\mid p^e-1$ , set $ \beta ={(p^e-1)}/{(p^{e-l}+1)}$ . Let $ \mathcal {C}=\bigoplus _{i=1}^{4}\gamma _i\mathcal {C}_i$ be an $ [n,k,d] $ linear code over $ \mathcal {R} $ , where the $ \mathcal {C}_i$ are the component codes over $ \mathbb {F}_q $ with generator matrices $ G_i$ . Let $ P_i=G_i{(F^{e-l}( G_i)) }^T $ and $ 0\leq t_i\leq k_i-1 $ be an integer such that $ \det ((P_i)_{S_i})=0 $ for any $S_i\subseteq \{1,2,\ldots , k_i\} $ with $ 0\leq |S_i|\leq t_i $ and assume there exist $ R_i\subseteq \{1,2,\ldots , k_i\} $ with cardinality $ t_i+1 $ such that $ \det ((P_i)_{R_i}) \neq 0$ . Suppose $\alpha \in \mathcal {R}^n$ such that $ \alpha _{ji}\in \mathbb {F}_q\setminus (\mathbb {F}_q^*)^{\,\beta }$ for all $ j\in R_i $ and $ \alpha _{ji}\in (\mathbb {F}_q^*)^{\,\beta } $ for all $ j\in \{1,2,\ldots , n\}\setminus R_i,$ for $ i=1,2,3,4$ . Then, $ \mathcal {C}^{\alpha } $ is an l-Galois LCD code over $ \mathcal {R} $ .

Proof. Since $\mathcal {C}^{\alpha }=\bigoplus _{i=1}^{4}\gamma _i \mathcal {C}_i^{\alpha _i'}, \ \text {where} \ \alpha _i'=(\alpha _{1i},\alpha _{2i},\ldots ,\alpha _{ni})\in \mathbb {F}_q^n $ and the $\mathcal {C}_i^{\alpha _i'}$ are linear codes over $\mathbb {F}_q$ with generator matrices $ G_i^{\alpha _i'}$ , by Theorem 4.5, the $\mathcal {C}_i^{\alpha _i'}$ are $ l $ -Galois LCD codes over $\mathbb {F}_q$ for $i=1,2,3,4$ . Therefore, by Theorem 3.2, $\mathcal {C}^{\alpha }$ is an $ l $ -Galois LCD code over $\mathcal {R}$ .

The following corollary shows the existence of $\alpha $ for which $\mathcal {C}^{\alpha }$ is an l-Galois LCD code equivalent to the linear code $\mathcal {C}$ over $\mathcal {R}$ .

Corollary 4.7. Let $ \mathbb {F}_q \ (q=p^e)$ be a finite field. For $0<l<e$ and $p^{e-l}+1\mid p^e-1$ , set $\beta ={(p^e-1)}/{(p^{e-l}+1)}$ ( $ \beta>1 $ ). Let $\mathcal {C}$ be an $ [n,k,d] $ linear code over the ring $ \mathcal {R} $ . Then, $ \mathcal {C}^{\alpha } $ is an $ [n,k,d]\ l $ -Galois LCD code over the ring $ \mathcal {R} $ for some $\alpha =(\alpha _1,\alpha _2,\ldots , \alpha _n)\in \mathcal {R}^n$ with $ \alpha _j\neq 0 $ for $ 1\leq j\leq n $ .

Proof. Let $ \mathcal {C}=\bigoplus _{i=1}^4\gamma _i\mathcal {C}_i $ be a linear code over $ \mathcal {R} $ . Take $ \alpha =(\alpha _1,\alpha _2,\ldots , \alpha _n) \in \mathcal {R}^n$ , where $ \alpha _j= \gamma _1+\gamma _2 +\gamma _3+\gamma _4$ for $ 1\leq j\leq n $ , if $\mathcal {C}$ is $ l $ -Galois LCD code over $ \mathcal {R} $ . Then, $ \mathcal {C}^{\alpha }=\mathcal {C} $ , which is an $ l $ -Galois LCD code over $ \mathcal {R} $ .

If $\mathcal {C}$ is not an $ l $ -Galois LCD code, then by Theorem 3.2, $\mathcal {C}_i $ is not an $ l $ -Galois LCD code for some $ 1\leq i\leq 4$ . If $ G_i $ is the generator matrix for $ \mathcal {C}_i $ , then $ {\det (G_i(F^{e-l}(G_i))^T)=0 }$ . Let $ P_i=G_i(F^{e-l}(G_i))^T $ . Then there exists an integer $ t_i\geq 0 $ and $ R_i \subseteq \{1,2,\ldots , k_i\} $ with cardinality $ | R_i| =t_i+1$ such that $ \det ((P_i)_{R_i})\neq 0 $ and $ \det ((P_i)_{S_i})=0 $ for any $ S_i \subseteq \{1,2,\ldots , k_i\} $ with cardinality $ 0\leq |S_i| \leq t_i $ . Also, since $\beta>1$ , $ \mathbb {F}_q^*\setminus (\mathbb {F}_q^*)^\beta \neq \emptyset $ . Choose $ \alpha _i'=(\alpha _{1i},\alpha _{2i},\ldots ,\alpha _{ni}) \in \mathbb {F}_q^n $ such that $ \alpha _{ji} \in \mathbb {F}_q^*\setminus (\mathbb {F}_q^*)^\beta $ for $ j\in R_i $ and $ \alpha _{ji}=1 $ for $ j\in \{1,2,\ldots , k_i\}\setminus R_i$ . By Theorem 4.5, $ \mathcal {C}_i^{\alpha _i'} $ is $ l $ -Galois LCD code over $\mathbb {F}_q$ . Take $ \alpha =\sum _{m=1}^4 \gamma _m \alpha _m' \in \mathcal {R}^n $ , where $ \alpha _m'=\alpha _i' $ for $ m=i $ and $ \alpha _m'=(1,1,\ldots ,1) $ for $ m\neq i $ . By Theorem 4.6, $ \mathcal {C}^{\alpha } $ is an $ [n,k,d]\ l $ -Galois LCD code over the ring $ \mathcal {R} $ .

5 MDS codes over $ \mathbf {\mathcal {R}} $

For a linear code $\mathcal {C}$ over the ring $\mathcal {R}$ with parameters $[n,k,d]$ , we have $|\mathcal {C}|\leq |\mathcal {R}|^{n-d+1}$ and so $d\leq n-\log _{|\mathcal {R}|}|\mathcal {C}|+1$ , the Singleton bound on the ring $\mathcal {R}$ . Since $|\mathcal {R}|=q^4$ and $|\mathcal {C}|=q^k$ , where $k=\sum _{i=1}^{4}k_i$ , the Singleton bound is $d\leq n-\tfrac 14\sum _{i=1}^{4}k_i+1$ . A code which attains the Singleton bound is called an MDS code. We have the following result for an MDS code over a finite field $\mathbb {F}_q$ .

Lemma 5.1 [11].

If C is a linear code over $\mathbb {F}_q$ , then the following are equivalent:

1. (1) C is an MDS code over $\mathbb {F}_q$ ;

2. (2) $C^{\perp }$ is an MDS code over $\mathbb {F}_q$ ;

3. (3) $C^{\perp _l}$ is an MDS code over $\mathbb {F}_q.$

The following theorem shows that a linear code is an MDS code if and only if its Euclidean ( $ l $ -Galois) dual is an MDS code over the ring $ \mathcal {R}$ .

Theorem 5.2. Let $ \mathcal {C}= \bigoplus _{i=1}^4\gamma _i \mathcal {C}_i $ be a linear code over the ring $\mathcal {R}$ , where the $ \mathcal {C}_i $ are the component codes over the finite field $ \mathbb {F}_q $ .

1. (1) $\mathcal {C}$ is an MDS code over the ring $\mathcal {R}$ if and only if the $\mathcal {C}_i$ are MDS codes over $\mathbb {F}_q$ with the same parameters for each $i=1,2,3,4$ .

2. (2) $\mathcal {C}$ is an MDS code over the ring $\mathcal {R}$ if and only if $\mathcal {C}^\perp $ is an MDS code over $\mathcal {R}$ .

3. (3) $\mathcal {C}$ is an MDS code over the ring $\mathcal {R}$ if and only if $\mathcal {C}^{\perp _l}$ is an MDS code over $\mathcal {R}$ .

Proof. (1) Suppose $\mathcal {C}$ is an MDS code over the ring $\mathcal {R}$ with parameters $[n,k,d]$ , where $4d=4n-\sum _{i=1}^{4}k_i+4$ . Since $d=\min _{1\leq i\leq 4}\{d_i\}$ , where $ d_i=d_H(\mathcal {C}_i) $ , it follows that $d=d_j$ for some $j=1,2,3,4$ . This implies $4d_j=4n-\sum _{i=1}^{4}k_i+4$ . Now $d_i\leq n-k_i+1$ for $i=1,2,3,4$ and so $\sum _{i=1}^{4}d_i\leq 4n-\sum _{i=1}^{4}k_i+4=4d_j$ . Since $d_j$ is the minimum of the $d_i$ for $i=1,2,3,4$ , we have $4d_j\leq \sum _{i=1}^{4}d_i$ . It follows that $4d_j=\sum _{i=1}^{4}d_i$ which is only possible when $d_1=d_2=d_3=d_4$ . Hence, the $\mathcal {C}_i$ are MDS codes over $\mathbb {F}_q$ with the same parameters.

Conversely, if the $\mathcal {C}_i$ are MDS codes with the same parameters, that is, $d_1=d_2=d_3=d_4$ and $d_i=n-k_i+1$ , then $4d_i=4n-\sum _{i=1}^{4}k_i+4$ for $i=1,2,3,4$ . Since $d=\min _{1\leq i\leq 4}\{d_i\}$ , this implies $4d=4n-\sum _{i=1}^{4}k_i+4$ . Hence, $\mathcal {C}$ is an MDS code.

(2) Let $\mathcal {C}$ be an MDS code over the ring $\mathcal {R}$ . By part (1), the $\mathcal {C}_i$ are MDS codes over $\mathbb {F}_q$ having the same parameters for each $i=1,2,3,4$ . Hence, the $\mathcal {C}_i^{\perp }$ are MDS codes over $\mathbb {F}_q$ with the same parameters for each $i=1,2,3,4$ . This implies that $\mathcal {C}^{\perp }$ is an MDS code over the ring $\mathcal {R}$ . A similar argument can be made for the converse.

(3) Let $\mathcal {C}$ be an MDS code over the ring $\mathcal {R}$ . Then, $\mathcal {C}^{p^{(e-l)}}$ is also an MDS code over the ring $\mathcal {R}$ . By Lemma 3.1, $\mathcal {C}^{\perp _l}=(\mathcal {C}^{p^{(e-l)}}) ^\perp $ , and hence $\mathcal {C}^{\perp _l}$ is an MDS code over $\mathcal {R}$ . Conversely, if $\mathcal {C}^{\perp _l}$ is an MDS code over $\mathcal {R}$ , it follows that $ \mathcal {C}^{p^{(e-l)}} $ is an MDS code. Hence, $\mathcal {\mathcal {C}}$ is an MDS code over $\mathcal {R}$ .

Remark 5.3. Result (1) in the above theorem over the ring $\mathbb {Z}_4+u\mathbb {Z}_4+v\mathbb {Z}_4+uv\mathbb {Z}_4$ is proved in [10].