Construction for both self-dual codes and LCD codes

08/28/2021
by   Keita Ishizuka, et al.
Tohoku University
0

From a given [n, k] code C, we give a method for constructing many [n, k] codes C' such that the hull dimensions of C and C' are identical. This method can be applied to constructions of both self-dual codes and linear complementary dual codes (LCD codes for short). Using the method, we construct 661 new inequivalent extremal doubly even [56, 28, 12] codes. Furthermore, constructing LCD codes by the method, we improve some of the previously known lower bounds on the largest minimum weights of binary LCD codes of length n=26,28 ≤ n ≤ 40.

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1 Introduction

Let be the finite field of order , where is a prime power. An code over is said to be a self-dual code if , where denotes the dual code of . A code is said to be doubly even if all codewords have weights divisible by four. Mallows and Sloane [22] proved that the minimum weight of a binary doubly even self-dual code of length is upper bounded by . A binary doubly even self-dual code meeting the bound is called extremal. An code over is said to be an LCD code if , where

denotes the zero vector of length

. The concept of LCD codes was invented by Massey [23]. A binary LCD code is said to be optimal if it has the largest minimum weight among all binary LCD codes.

Although the definitions say that self-dual codes and LCD codes are quite different classes of codes, codes of both classes have similar properties. For example, it is known that both self-dual codes and LCD codes are characterized by their generator matrices. Furthermore, self-dual codes are codes with maximal hull dimension and LCD codes are codes with minimal hull dimension, where the hull of a code is defined as . Recently Harada [16] gave a method for constructing LCD codes modifying known methods for self-dual codes in [14, Theorem 2.2] and [18, Theorem 2.2] and constructed optimal binary LCD codes.

In this paper, we give a method for constructing many code from a given code such that . This method can be applied to constructions of both self-dual codes and LCD codes. It is shown that the method is a generalized version of [14, Theorem 2.2][16, Theorem 3.3] and [18, Theorem 2.2]. As an application, we construct new inequivalent extremal doubly even codes. Furthermore, constructing LCD codes by the method, we improve some of the previously known lower bounds on the largest minimum weights of binary LCD codes of length .

This paper is organized as follows: In Section 2, we recall some basic results on self-dual codes, LCD codes and hulls of codes. In Section 3, we provide the construction method. Furthermore, in Section 4, we state basic properties of the construction method. In Section 5, we construct new inequivalent extremal doubly even from six bordered double circulant doubly even codes. In Section 6, we improve some of the largest minimum weights among all binary LCD codes with length , which were recently studied by Bouyuklieva [7] and Harada [16]. All computations in this paper were performed in MAGMA [6].

2 Preliminaries

Let be the finite field of order , where is a prime power and let be the vector space of all -tuples over . A -dimensional subspace of is said to be an code over . Especially, codes over are said to be binary codes. Let be an code over . The parameters , are said to be the length, the dimension of respectively. A vector in is said to be a codeword. The weight of is defined as . The minimum weight of is defined as . If the minimum weight of equals to , then is said to be an code over . A code is said to be an even code if all codewords have even weights. Also, a code is said to be a doubly even code if all codewords have weights divisible by four. Two codes over are equivalent if there exists a monomial matrix such that . The equivalence of two codes is denoted by . A generator matrix of a code is any matrix whose rows form a basis of .

The dual code of an code over is defined as , where is the standard inner product. If , then is said to be a self-orthogonal code. If , then is said to be a self-dual code. A binary self-dual code is doubly even if and only if , where denotes the length of . Mallows and Sloane [22] proved that the minimum weight of a binary doubly even self-dual code of length is upper bounded by . A binary doubly even self-dual code meeting the bound is called extremal.

Lemma 2.1 ([20, Theorem 1.4.8]).

Let be a binary code. Then the following holds:

  1. If is a self-orthogonal code and has a generator matrix each of whose rows has weight divisible by four, then is doubly even.

  2. If is doubly even, then is a self-orthogonal code.

A pure double circulant code has a generator matrix of the form and a bordered double circulant code has a generator matrix of the form

(1)

where

denotes the identity matrix of order

and is a circulant matrix. These two families of codes are collectively called double circulant codes. Harada, Gulliver and Kaneta [17] showed that there exist exactly nine inequivalent extremal double circulant doubly even codes and all of them are bordered double circulant codes. In Section 5, we construct extremal doubly even self-dual codes from six inequivalent extremal double circulant doubly even codes . Generator matrices of are of the form (1) with first rows

respectively.

An code over is said to be an LCD code if . The concept of LCD codes was invented by Massey [23]. LCD codes have been applied in data storage, communication systems and cryptography. For example, it is known that binary LCD codes can be used against side-channel attacks and fault injection attacks [9]. A binary LCD code is said to be optimal if it has the largest minimum weight among all binary LCD codes. Massey [23] gave the following characterization of LCD codes.

Theorem 2.2 (Massey [23]).

Let be an code over and let be a generator matrix of . Then is an LCD code if and only if the matrix is nonsingular.

The hull of a code is defined as . By definition, it follows that self-dual codes are codes with maximal hull dimension and LCD codes are codes with minimal hull dimension.

Lemma 2.3 ([13, Proposition 3.1]).

Let be an code over with generator matrix . Then

3 Construction method

Let be an code over with generator matrix and let . We denote by the -th row of . Define an matrix , where the -th row is defined as follows:

We denote by the code with generator matrix .

Remark 3.1.

With the above notation, suppose that or . Then it holds that . Hereafter, we assume that and .

Theorem 3.2.

Let be an code over with generator matrix and let . Suppose that . Then .

Proof.

We denote by the -th rows of respectively. It holds that

Therefore it follows that

By Lemma 2.3, the result follows. ∎

Corollary 3.3.

Let be an code over with generator matrix and let . Suppose that . Then is a self-orthogonal code if and only if is a self-orthogonal code.

Proof.

It holds that is a self-orthogonal code if and only if . The result follows from Theorem 3.2. ∎

Remark 3.4.

With the notation of Corollary 3.3, suppose that is a binary self-dual code with length . Then it follows that if has an even weight. Let be the -th rows of respectively. Since is an even code, for all . Therefore we obtain the following:

which shows that is identical to , the first case of [14, Theorem 2.2]. Therefore Corollary 3.3 is a generalized version of the first case of [14, Theorem 2.2].

Corollary 3.5.

Let be an code over with generator matrix and let . Suppose that . Then is an LCD code if and only if is an LCD code.

Proof.

It holds that is an LCD code if and only if . The result follows from Theorem 3.2. ∎

Remark 3.6.

With the notation of Corollary 3.5, suppose that is a binary even LCD code and is even. Then it follows that if has an even weight. Let be the -th rows of respectively. Since is an even code, for all . Therefore we obtain the following:

which shows that is identical to in [16, Theorem 3.3]. Therefore Corollary 3.5 is a generalized version of [16, Theorem 3.3].

Lemma 3.7 ([20, Theorem 1.4.3]).

Let . Then , where .

Lemma 3.8.

Let be a binary code with generator matrix and let . Suppose that and . Then , where denote the -th rows of respectively.

Proof.

It holds that

where we regard as vectors in . Therefore, by Lemma 3.7, it follows that

This completes the proof. ∎

Theorem 3.9.

Let be a binary code with generator matrix and let . Suppose that and . Then is a doubly even code if and only if is a doubly even code.

Proof.

We denote by the -th rows of respectively. Suppose that is a doubly even code. Then, by Corollary 3.3 and the second part of Lemma 2.1, is a self-orthogonal code. Furthermore, by Lemma 3.8, follows for all . Therefore it holds that is a doubly even code by the first part of Lemma 2.1 By the same argument, the converse holds. ∎

Remark 3.10.

With the notation of Theorem 3.9, suppose that is a binary doubly even self-dual code. Then it follows that and if has a weight divisible by four. Let be the -th rows of respectively. Since is doubly even, for all . Therefore we obtain the following:

which shows that is identical to in [18, Theorem 2.2], where is even. Therefore Theorem 3.9 is a generalized version of [18, Theorem 2.2], where is even.

4 Basic properties

In this section, we state basic properties of the construction method given in Section 3. Let . Define the following matrix :

Lemma 4.1.

Let be an code over with generator matrix and let . Then it holds that

Proof.

For a matrix , we denote by the -entry of . By definition it follows that

On the other hand, it holds that

where denote the -th rows of respectively. Therefore it holds that

This completes the proof. ∎

Theorem 4.2.

Let be an code over with generator matrix and let . Suppose that . Then the following holds:

(2)
(3)
Proof.

For (2), it holds that

For (3), the following holds:

This completes the proof. ∎

In the following sections, we apply the construction method only to binary codes. However, for codes over , Theorem 4.2 reduces computations.

5 Extremal binary doubly even codes

In this section we are concerned only with binary codes. Therefore we omit the term “binary”. Bhargava, Young and Bhargava [5] constructed an extremal doubly even code. Yorgov [26] proved that there exist exactly inequivalent extremal doubly even codes with automorphisms of order . As stated in [15], one of the codes in [26] is equivalent to the code in [5]. Bussemarker and Tonchev [8] constructed extremal doubly even codes. As stated in [15], the first code among the six codes in [8] is equivalent to the code in [5]. Moreover, Kimura [21] showed that th and th codes in [8] are equivalent. Harada [14] constructed inequivalent extremal doubly even codes. Harada, Gulliver and Kaneta [17] showed that there exist exactly nine inequivalent extremal double circulant doubly even codes. Harada [15] constructed inequivalent extremal doubly even codes. This result is a generalization of [14] and for any code in [14] there exists a code in [15] such that . Yankov and Russeva [25] proved that there exist exactly inequivalent extremal doubly even codes having automorphisms of order . As stated in [25], one of the codes in [25] is equivalent to a code in [14]. Yankov and Lee [24] proved that there exist exactly inequivalent extremal codes having automorphisms of order . Therefore the number of previously known extremal doubly even codes is , as stated in [24, Proposition 7].

In [15], Harada applied [18, Theorem 2.2] to six inequivalent extremal double circulant doubly even codes . As stated earlier, Theorem 3.9 is a generalized version of [18, Theorem 2.2]. In this section, we apply Theorem 3.9 to and construct new inequivalent extremal doubly even codes. This illustrates the effectiveness of Theorem 3.9.

In order to illustrate our method, we consider the code as an example. Let denote the vector of length such that . Applying Theorem 3.9 to and all such that and , we constructed inequivalent extremal doubly even codes. Furthermore, applying Theorem 3.9 to and all such that and , we constructed inequivalent extremal doubly even codes respectively.

By the following method, we verified that the above codes are all inequivalent. For an extremal doubly even code , we define to be an matrix whose rows composed of codewords of with weight . Furthermore, for a positive integer , we define

Harada [15] showed that two extremal doubly even codes are inequivalent if the sequences constructed from are distinct. According to this result, we compared the sequence for the classification. Consequently we found no pair of codes whose sequences are identical. Therefore we verified that the number of inequivalent codes constructed from is . By the same method, we constructed inequivalent extremal doubly even codes from . In Table 1 we show the number of inequivalent codes constructed by this method. We denote the inequivalent codes constructed from by respectively. The in Corollary 3.9 for all codes we constructed can be obtained electronically from https://www.math.is.tohoku.ac.jp/~mharada/Ishizuka/56.txt.

total
Table 1: Inequivalent extremal doubly even codes

Comparing sequences , we found that there exist four pairs of codes whose sequences are identical. By the Magma function IsIsomorphic, we verified that two codes of all the four pairs are equivalent. Therefore the number of inequivlent extremal doubly even codes constructed as above is .

Finally, we verified that the codes are inequivalent to any of the previously known extremal doubly even codes as follows: By the Magma function AutomorphismGroup, we verified that the codes have automorphism groups of order . Consequently it follows that the codes are inequivalent to any of the codes in [24][25][26]. Furthermore we verified that all the codes except have sequences different from that of any code in [5][8][15][17][26]. The sequence of is identical to that of the th code constructed from in [15]. However we verified by the Magma function IsIsomorphic that the two codes are inequivalent. Consequently it follows that the codes are inequivalent to any of the codes in [5][8][15][17][26]. As stated in the beginning of this section, the number of the previously known inequivalent doubly even codes is . Therefore we have Proposition 5.1.

Proposition 5.1.

There exist at least inequivalent extremal doubly even codes.

6 Optimal binary LCD codes of length

In this section we are concerned only with binary codes. Therefore we omit the term “binary”. Let denote the largest minimum weight among all LCD code. Galvez, Kim, Lee, Roe and Won [11], Harada and Saito [19], Araya and Harada [1] determined the exact value of for respectively. Bouyuklieva [7] determined the exact value of for and gave for . Galvez, Kim, Lee, Roe and Won [11], Harada and Saito [19], Araya and Harada [1], Araya, Harada and Saito [3] determined the exact value of for respectively. Also, Dougherty, Kim, Ozkaya, Sok and Solé [10], Araya and Harada [2], Araya, Harada and Saito [4] determined the exact value of for , , respectively. For all , Bouyuklieva [7] determined the exact value of for .

Recently Harada [16] constructed optimal LCD codes by [16, Theorem 3.3]. As stated earlier, Corollary 3.5 is a generalized version of [16, Theorem 3.3]. In this section, we apply Corollary 3.5 in order to improve some of the previously known lower bounds on for and . For a code , we denote by the punctured, the shortened codes of on a set of coordinates respectively. In this section, shortened codes, punctured codes were constructed by the Magma functions ShortenCode, PunctureCode respectively.

In order to obtain lower bounds, we use the following method: First, by the Magma function BestKnownLinearCode, we obtained a code , a code and a code . Generator matrices of these codes can be obtained electronically from https://www.math.is.tohoku.ac.jp/~mharada/Ishizuka/generator.txt. Then we verified that , , are an LCD code, an LCD code, an LCD code respectively, where the set of coordinates are given in Table 2. Define , , as in Figure 1. Then , , are generator matrices of the LCD , the LCD , the LCD codes respectively. Applying Corollary 3.5 to , , , we found an LCD code , an LCD code , an LCD code respectively. The vectors in Corollary 3.5 are listed in Table 3. Therefore we obtain Proposition 6.1.

Proposition 6.1.
  1. There exists an LCD code.

  2. There exists an LCD code.

  3. There exists an LCD code.

Table 2: for
Table 3: with

From the previously known results of described in the beginning of this section, we are concerned only with for and . Let denote the largest minimum weight among currently known codes. By the Magma function BestKnownLinearCode, one can construct an code for all and . In addition, by considering shortened codes and punctured codes of codes, we found LCD codes for

(4)

Consequently we obtain Proposition 6.2.

Proposition 6.2.

There exists an optimal LCD code for listed in (4).

By a method similar to that given in the above, we found LCD codes and LCD codes for