## 1 Introduction

Let be a finite field of order , where and is a prime. Recently, Fan and Zhang [12] generalize the Euclidean inner product and the Hermitian inner product to the so-called -Galois form (or -Galois inner product), where . The -Galois dual codes, and the -Galois self-dual constacyclic codes over finite fields are studied. In particular, necessary and sufficient conditions for the existence of -Galois self-dual and isometrically Galois self-dual constacyclic codes are obtained. As consequences, some results on self-dual, iso-dual and Hermitian self-dual constacyclic codes are derived.

Linear complementary-dual (LCD for short) codes are linear codes that intersect with their duals trivially. They were first studied by Massey [23] who showed that these codes are optimal for the two-user binary adder channel (BAC) and that they are asymptotically good. Sendrier [29] showed that these codes meet the Gilbert-Varshamov bound. In ([28],[29],[30],[32], [31]), the authors also studied the hulls of linear codes, and tried to find permutations between two equivalent codes, which has an application to code-based public key cryptosystems. Carlet and Guilley gave some applications of LCD codes in side-channel attacks and fault non-invasive attacks ([4],[6],[7]). LCD codes also can be used for constructions of lattices [17]. Optimal and MDS codes that are LCD are studied in many papers (see [2],[8],[10],[12],[16],[20],[22],[23],[27],[19]).

Motivated by the previous work, we study the Galois hulls of linear codes over finite fields. The -Galois hull of a linear code over a finite field is defined by , where , is a prime, and . The classical LCD code is a linear code with , and the Hermitian LCD code is a code with , where is even.

Construction of codes is an interesting research field in coding theory. The matrix product code is a new code constructed from the codes of same length and an matrix over a finite field . These codes were first proposed and studied in [5]. There are many papers focusing on its algebraic structure, different distance structures, and decoding algorithm (see [14],[15],[24],[1],[11]).

This paper is organized as follows. Section 2 gives some preliminaries. In Section 3, a characterization of the dimension of the -Galois hull of a linear code is provided. As a corollary, we obtain a necessary and sufficient condition for a linear code to be an -Galois LCD code. In Section 4, we first show that the dimension of any -Galois hull of a linear code is invariant under permutation equivalence for . For ternary codes, the dimension of the -Galois hull is also invariant under monomial equivalence. Then we show that every linear code over is monomial equivalent to an -Galois LCD code in the case of . We conclude that if there exists an linear code over with , then there exists an -Galois LCD code with the same parameters. In Section 5, we study the structure and the dimension of -Galois hull of matrix product codes.

## 2 Preliminaries

Throughout this paper, denotes a finite field of order , where is a prime, is a positive integer. By we denote the multiplicative group of . Let be the

dimensional vector space over

. Any subspace of is called a linear code of length over . We assume that all codes are linear in this paper.Let be the symmetric group on the set . For all and , acts on in the following way.

In [12], Fan and Zhang introduced the following concept.

###### Definition 2.1.

Assume the notations given above. For each integer with , let

Then the form is called the -Galois form on , or -Galois inner product.

It is easy to see that is just the usual Euclidean inner product. And, is the Hermitian inner product if is even. For any code over , the following code

is called the -Galois dual code of . If , then is said to be -Galois self-orthogonal. Moreover, is said to be -Galois self-dual if .

Note that is linear whenever is linear or not. In particular, ( for short) is just the Euclidean dual code of , and ( for short) is just the Hermitian dual code of if is even.

Let , be the Frobenius automorphism of . For any , and any matrix over , set and .

The following proposition is easily obtained.

###### Proposition 2.2.

Assume the notations given above. Then for any ,

(1)

(2) , for any . In particular, , and if is even.

###### Proof.

The two statements follow immediately from the identity ∎

###### Definition 2.3.

Let be a linear code over . The -Galois hull of is defined by . If , then is called a linear code with -Galois complementary dual or an -Galois LCD code. If , then is called an -Galois self-orthogonal linear code.

###### Remark 2.4.

Note that when and , the code is the classical LCD code. When is even and , the code is the Hermitian LCD code.

A monomial matrix is a square matrix such that in every row (and in every column) there is exactly one nonzero element. It is easy to see that any monomial matrix is a product of a permutation matrix and an invertible diagonal matrix. In particular, a permutation matrix is a special monomial matrix. Two linear codes and of length over are monomial equivalent, if there is an monomial matrix of size such that . If is a permutation matrix, then and are called permutation equivalent.

## 3 The -Galois hull of linear codes

In this section, we give a characterization for the -Galois hull of any linear code over . We have the following theorem.

###### Theorem 3.1.

Let be an linear code over with a generator matrix . Let be the dimension of the -Galois hull of , and let . Then there exists a generator matrix of such that

where and are respectively zero matrices of sizes and , and the rank of is . Furthermore, the rank of is for any generator matrix of .

###### Proof.

Let be a basis of . We can extend to a basis of . Let be the matrix such that its th row is , where . Then is a generator matrix of and is a matrix. The element at the -entry of is . Note that if , since for all and for all . Therefore, has the form as stated in the theorem.

Now we show that . Obviously, . Suppose . Then there exists a non-zero vector such that . Let where is the zero vector of length . Then we have

Since the map is an automorphism of , there exists a vector such that . Therefore,

This gives that , which implies . We also have

Hence since are linear independent. This is a contradiction. Hence, .

Let be an arbitrary generator matrix of , then there exists an invertible matrix such that . We have

Then , since the matrix and are invertible. We are done. ∎

The following corollary can be obtained immediately.

###### Corollary 3.2.

([25]) Let be an linear code over with a generator matrix . Let be the dimension of and . Then the code has a generator matrix such that

where are all zero matrices, and is an invertible matrix. Furthermore, the rank of is for every generator matrix of .

###### Proof.

###### Corollary 3.3.

Let be an linear code over with a generator matrix , where , and is even. Let be the dimension of and . Then has a generator matrix such that

where are all zero matrices, and is an invertible matrix. Furthermore, the rank of is for every generator matrix of .

###### Proof.

Take in Theorem 3.1, and note that . It is easy to verify that for any . Hence if and only if . The result then follows immediately. ∎

###### Remark 3.4.

The following example shows that if , or , where is even, then the matrix may not be . For example, let and be a linear code of length with a generator matrix . Then .

When or for a linear code , then the following two corollaries are straightforward.

###### Corollary 3.5.

([21]) Let be an linear code over with a generator matrix . Then is -Galois LCD code if and only if is nonsingular.

###### Corollary 3.6.

Let be an linear code over with a generator matrix . Then is an -Galois self-orthogonal code if and only if .

In particular, we have

###### Corollary 3.7.

Let be an linear code over with a generator matrix and a parity check matrix . Then is an -Galois self-dual code if and only if both and are .

###### Proof.

Since is a parity check matrix of , is a generator matrix of . Note that if and only if , and if and only if if and only if . This finishes the proof. ∎

## 4 The existence of -Galois LCD codes

LCD codes over finite fields are an important class of linear codes. They have many applications in coding theory and cryptography, especially in designing decoding algorithm. In this section, we focus on the equivalence of -Galois LCD codes.

###### Lemma 4.1.

Let be an linear code of length over , be a permutation, and . Then

(1) , for any .

(2) .

###### Proof.

(1) Let , then and . Hence

(2) Let , then . For any , there exists a codeword such that . We have

This implies that and hence . Since

we get . ∎

###### Proposition 4.2.

The dimension of the -Galois hull of a linear code is invariant under permutation equivalence.

###### Proof.

Let be an linear code of length over and be an arbitrary permutation. By Lemma 4.1, we have

Because , this finishes the proof. ∎

###### Remark 4.3.

If , the dimension of the -Galois hull of a linear code is invariant under monomial equivalence. In fact, suppose that , by Proposition 4.2, we only need to prove the case when is an invertible diagonal matrix. Let and be the generator matrices of and respectively, then . Since , we have and

, the identity matrix. Hence we have

It follows that .

In order to prove the main result in this section, we need the following proposition. This proposition is known (for example, see [25]), we provide an alternative proof here.

###### Proposition 4.4.

Let be a nonzero polynomial of such that the degree of with respect to is at most for all , where . Then there exists a vector such that .

###### Proof.

We prove this proposition by induction on . If , then by assumption the degree of is at most . Note that the number of roots of over the finite field is less or equal to the degree of . Therefore, there exists a vector such that .

Now assume . We can further assume that the degree of with respect to is greater than or equal to . Otherwise , then there exists a vector such that by the inductive hypothesis. Therefore we can assume that

where and is a nonzero polynomial. Hence, there exists such that . Let and , then there is an element such that by the result of the previous argument. Hence there exists such that . ∎

###### Proposition 4.5.

Let and be two nonzero polynomials of such that the degree of with respect to is at most for all . Let . Then there exists a vector such that .

###### Proof.

By Proposition 4.4, there exists a vector such that . Hence there exists a vector such that . ∎

By the proposition above, we can easily get the following two corollaries.

###### Corollary 4.6.

Let be a nonzero polynomial of such that the degree of with respect to is at most for all . Then there exists an such that .

###### Proof.

Let and . Then . We are done. ∎

###### Corollary 4.7.

Let and . Let be a nonzero polynomial of such that the degree of with respect to is at most for all . Then there exists a vector such that .

###### Proof.

Let and . Then . The result then follows immediately. ∎

###### Theorem 4.8.

Let be an linear code over , where and . Then is monomial equivalent to an -Galois LCD code.

###### Proof.

Let be an linear code over with . Without loss of generality, we may assume that has a generator matrix of the standard form . Let . Now we define a -variable polynomial as follows:

It is easy to verify that is a nonzero polynomial with the variables and the degree of with respect to is for all . In the following, we show that . Note that , we have

If and , then , and . This implies that . Therefore, .

If and , then , hence .

If and , then , and we get . Hence, .

Therefore, by Corollary 4.6, there exists a vector such that .

Now let be the matrix, where is an diagonal matrix with the form . Let be the code with the generator matrix . Then is monomial equivalent to . And we have

Therefore, is an -Galois LCD code by Corollary 3.5. ∎

When , we have the following corollary.

###### Corollary 4.9.

In fact, this corollary is also true when (see [9]). When is even, and , we get the following corollary.

###### Corollary 4.10.

###### Remark 4.11.

When , the corollary above is not right in general. In fact, if we let be the finite field of order , where . Let , and let be a generator matrix of . Then and so is not an Hermitian LCD code. For any , let be a generater matrix of the code . Then

Then for all , we have . Hence any code that is monomial equivalent with is not an Hermitian LCD code.

###### Theorem 4.12.

Let be a linear code over , where and . Then there exists a such that is an -Galois LCD code.

###### Proof.

Let be an linear code over . Without loss of generality, we may assume that has a generator matrix of the form . Let and be the the generator matrix of the code , where is defined in Theorem 4.8. Let . Now we define . Hence is a polynomial with the variables and the degree of with respect to is for all . We know that and . The leading term of with respect to the total degree of lex order is . So is a nonzero polynomial. Therefore for some by Corollary 4.7. Hence is an -Galois LCD code by this choice of , and Corollary 3.5 because . ∎

## 5 An application to matrix product codes

In this section, we apply the results obtained in Section 4 to study the hull of matrix product codes over finite fields.

Let be an matrix, be an

matrix. The tensor product of the two matrices is defined by

. The following properties of the tensor product of matrices are well-known.###### Lemma 5.1.

Let and . Then

(1) .

(2) .

###### Definition 5.2.

Let be an matrix over , and let be codes of length over . The matrix product code

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