## 1 Introduction

In this paper, let be a finite field with elements. The set of non-zero elements of is denoted by . For any , the conjugate of is defined as . A dimensional subspace of is called an linear code with minimum (Hamming) distance . Given a linear code of length over (resp. ), its Euclidean dual code (resp. Hermitian dual code) is denoted by (resp. ). The codes and are defined as follows

A linear code has complementary dual (or LCD code for short) over if . The Euclidean (resp. Hermitian) hull of a linear code is defined to be (resp. ). A linear code over is called a Euclidean (resp. Hermitian) LCD code if (). In the later of this paper, Euclidean LCD code is abbreviated to LCD code if no special stated.

In 1992, Massey first initiated LCD codes [1], and he also proved the existence of asymptotically good LCD codes. Sendrier showed that LCD codes meet the asymptotic Gilbert-Varshamov bound over the finite fields [2]. Yang and Maseey gave a necessary and sufficient condition for a cyclic code to be LCD over finite fields [3]. After that, there are many literatures on the construction of LCD codes over finite fields [4, 5, 6, 7, 8, 9]. What’s more there are many LCD MDS code have been constructed by some scholars in [10, 11, 12, 13]. Carlet et al. solved the problem of the existence of -ary LCD MDS codes for Euclidean case [12], they also introduced a general construction of LCD codes from any linear codes. Further more, they showed that any linear code over is equivalent to an Euclidean LCD code and any linear code over is equivalent to a Hermitian LCD code [13]. Sok et al. proved the existence of optimal LCD codes over large finite fields [14]. Liu et al. discussed the structure of LCD codes over finite chain rings[15].

Recently, many researchers have an interest in LCD codes over small finite fields[16, 17, 18, 19]. Galvez et al. gave bouds on the minimum distances of binary LCD codes with fixed lengths and dimensions on the dimensions of LCD codes with fixed lengths and minimum distances.[17]. Carlet et al. presented a new characterization of binary LCD codes in terms of their symplectic basis and solve a conjecture proposed by Galvez et al.[18]. Harada et al. studied binary LCD codes with the largest minimum weight among all binary LCD codes[19]. Inspired by these latter works, we consider the bounds on LCD codes over small finite fields.

In this paper, we give some background and recall some basic results in Section 2. In Section 3, we show that LCD codes are not equivalent to linear codes over small finite fields. In Sections 4 , the enumeration of binary optimal LCD codes is obtained. In Sections 5 and 6, we get the exact value of LD over and . In Section 7, we study the bound of LCD codes over .

## 2 Preliminaries

Two codes and in are called equivalent if for some permutation of and . For a matrix over finite field, denotes the transposed matrix of and denotes the conjugate of . We assume that det denotes the determinant of A, where is a square matrix. Hamming weight vector is the number of nonzero , and denoted by wt.

###### Lemma 2.1 (see [20]).

If is a generator matrix for the linear code , then is an Euclidean (resp. a Hermitian) LCD code if and only if, the matrix (resp. ) is nonsingular.

###### Lemma 2.2.

Let be a matrix. Let be the columns vectors of and , where is a permutation of . Then det=det.

###### Proof.

By the definition of , there are primary matrixes such that . We have

Then the proof is completed.∎

The combinatorial functions LD and LD has been introduced and studied by Dougherty et al. [22] and Galvez et al. [17]. The definitions of LD and LD as follows, we will use them frequently in the rest of this paper.

###### Definition 2.3.

LD.

###### Definition 2.4.

LD.

Let be an linear code over and the matrix be the generator matrix of . Then the size of is and rank. Let

where , for and .

Any code over , we have the following inequality.

###### Lemma 2.5.

, for .

###### Proof.

From the Griesmer Bound [21], for any -ary linear code, we have

we have . Hence

Therefore any LCD code must satisfy this inequality.∎

###### Lemma 2.6.

Let and are positive integers, , then LDLD

###### Proof.

Let G be a generator matrix of an LCD code over . Then with the generator matrix is an LCD code since det=det. Note that is an code. This completes the proof.∎

## 3 LCD codes are not equivalent to linear codes over small finite fields

In this section, we investigate the relationship between linear codes and LCD codes over small finite fields. In [20], Carlet et al. showed that an linear Euclidean LCD code over with exists if and only if there is an linear code over and an linear
Hermitian LCD code over with exists if and only if there is an linear code over . Now we proved that this result is not true in the small finite fields, such as , and .

###### Theorem 3.1.

Let be a linear code over with generator matrix and assume is not an Euclidean LCD code. Then is not equivalent to any Euclidean LCD codes over .

###### Proof.

Assume is equivalent to a linear code , where is a permutation of and . It is obvious that . Let and are the generator matrixes of and , respectively. It is easy to know det()=0, then det from Lemma 2.2. We show that linear code is not to be Euclidean LCD.∎

###### Theorem 3.2.

Let be a linear code over with generator matrix and assume is not an Euclidean LCD code. Then is not equivalent to any Euclidean LCD codes over .

###### Proof.

Assume is equivalent to a linear code , where is a permutation of and . Let , and are the generator matrixes of , and , respectively. The is obtained from by multiplying its th column by for , then we have by simple matrix operations. It is easy to know det()=0, then det from Lemma 2.2. We show that linear code is not to be Euclidean LCD.∎

###### Theorem 3.3.

Let be a linear code over with generator matrix and assume is not a Hermitian LCD code. Then is not equivalent to any Hermitian LCD codes over .

###### Proof.

Assume is equivalent to a linear code , where is a permutation of and . Let , and are the generator matrixes of , and , respectively. The is obtained from by multiplying its th column by for and , then we have by simple matrix operations. It is easy to know det()=0, then det from Lemma 2.2. We show that linear code is not to be Hermitian LCD. ∎

Thus in the later section, we only consider LCD codes over , and .

## 4 The enumeration of binary optimal LCD codes

In this section we consider binary codes. Recently, Galvez et al. [22] obtain the exact values of LD for and arbitrary . By Theorem 1 in [22], we know that there exist LCD codes with LD only for . An linear code is optimal if the minimum distance achieve the Gresmer Bound. In this section, we will give the enumeration of binary optimal LCD codes for , where .

An linear code over with generator matrix , Let

Let , , then

Let for , then .

The following definition will be frequently in this section.

###### Definition 4.1.

, for .

From this definition and notation given above, we have . It is easy to know

We also have

Based on the notation given above, we can obtain the following theorems.

###### Theorem 4.2.

Up to equivalence, the number of binary optimal LCD codes is , for .

###### Proof.

(i) Let , i.e., , for some positive integer . Let the code with generator matrix , let min, then , where . If , where . Note that the matrix of those codes always satisfy det. Therefor those codes are LCD code. But the code generator by is equivalent to the code generator by . If , where . Note that the matrix of those codes always satisfy det. Therefor those codes are not LCD code. Hence, there are only two binary optimal LCD codes.

(ii) Let , i.e., , for some positive integer . The proof is similar to (i), we omit detail here.We obtain , where . If , where . Note that the matrix of those codes always satisfy det. Therefor those codes are LCD code. But the code generator by is equivalent to the code generator by . If , where . Note that the matrix of those codes always satisfy det. Therefor those codes are not LCD code. Hence, there are only two binary optimal LCD codes.

(iii) Let , i.e., , for some positive integer . The proof is similar to (i), we omit detail here. We obtain . If , note that the matrix of those codes always satisfy det. Therefor those codes are LCD code. But the code generator by is equivalent to the code generator by . Hence, there are only two binary optimal LCD codes.∎

###### Theorem 4.3.

Up to equivalence, the number of binary optimal LCD codes is , for .

###### Proof.

Let , i.e., , for some positive integer . Let the code with generator matrix , let min, then , where . If , where . Note that the generator matrix of this code satisfy i.e., det. Therefor this code is an LCD code. If , where . Note that the matrix of those codes always satisfy det. Therefor those codes are not LCD code. Hence, there are only one binary optimal LCD code. ∎

###### Theorem 4.4.

The let be a positive integer, then

(1) Suppose that is even and . If , then LK=1.

###### Proof.

(1) Let be an LCD code over with generator matrix G. If , there are an LCD code by Theorem 3.4 in [18]. From the Griesmer bound, we have . This is imply that there is no code when . That is, there is no such an LCD code.

If , because the minimum distance is odd, thus we get . There is LCD code by Lemma 2.1.

(2) Let be an LCD code over with generator matrix G. If , there are an LCD code by Theorem 3.4 in [18]. From the Griesmer bound, we have . Since LD from Theorem 2 in [17]. Thus there is no . This is imply that there is no code when . That is, there is no such an LCD code.

If , because the minimum distance is odd, thus we get . There is LCD code by Lemma 2.1. ∎

## 5 The exact value of LD over

In this section, we consider ternary codes and give the exact value of LD over for and arbitrary .

An linear code over with generator matrix , Let

Let , , then

Let for , then .

The following definition will be frequently in this section.

###### Definition 5.1.

, for .

From this definition and notation given above, we have . It is easy to know

We also have

Based on the notation given above, we can obtain the following theorems.

###### Theorem 5.2.

for

###### Proof.

From the Lemma 2.5, let and , we get this inequality.∎

###### Theorem 5.3.

Let . Then for .

###### Proof.

Let linear code generate by give above, we only need to show the existence of LCD code with minimum distance .

(i) Let , i.e., , for some positive integer . If is an odd integer, let the code with generator matrix , let and . Note that this code has minimum distance and i.e., det. Therefor this code is an LCD code. If is an even integer, let the code with generator matrix , let , and , Note that this code has minimum distance and i.e., det. Therefor this code is an LCD code.

(ii) Let , i.e., , for some positive integer . If is an odd integer, let the code with generator matrix , let and , Note that this code has minimum distance and i.e., det. Therefor this code is an LCD code. If is an even integer, let the code with generator matrix , let and . Note that this code has minimum distance and i.e., det. Therefor this code is an LCD code.∎

###### Theorem 5.4.

Let . Then for .

###### Proof.

Let linear code generate by give above, we will show there is no LCD code with minimum distance .

(i) Let , i.e., , for some positive integer . Let the code with generator matrix , let min, then, and . Note that i.e., det. Therefor those codes are not LCD codes. Furthermore, if is an odd integer, let the code with generator matrix , let and . Note that this code has minimum distance and i.e., det. Therefor this code is an LCD code. If is an even integer, let the code with generator matrix , let , and . Note that this code has minimum distance and i.e., det. Therefor this code is an LCD code.

(ii) Let , i.e., , for some positive integer . Let the code with generator matrix , let min, then, . (1). Let , then , if , we have , then , we obtain , which is impossible. If , we have , then , we obtain , which is also impossible. (2). Let , , then , , if , we have , then , we obtain , which is impossible. If , we have , then . Note that i.e., det. Therefor this code is not an LCD code. (3). Let , . It is similar to (2). We get i.e., det. Therefor this code is not an LCD code. (4). Let , then , if , we have , then . Note that i.e., det. Therefor this code is not an LCD code. If , we have , then . Note that

Comments

There are no comments yet.