Quaternary Hermitian linear complementary dual codes

04/16/2019
by   Makoto Araya, et al.
Tohoku University
0

The largest minimum weights among quaternary Hermitian linear complementary dual codes are known for dimension 2. In this paper, we give some conditions for the nonexistence of quaternary Hermitian linear complementary dual codes with large minimum weights. As an application, we completely determine the largest minimum weights for dimension 3, by using a classification of some quaternary codes. In addition, for a positive integer s, a maximal entanglement entanglement-assisted quantum [[21s+5,3,16s+3;21s+2]] codes is constructed for the first time from a quaternary Hermitian linear complementary dual [26,3,19] code.

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1 Introduction

Linear complementary dual (LCD for short) codes are linear codes that intersect with their dual trivially. LCD codes were introduced by Massey [14] and gave an optimum linear coding solution for the two user binary adder channel. Recently, much work has been done concerning LCD codes for both theoretical and practical reasons (see e.g. [1], [5], [6], [8], [11], [12] and the references given therein). For example, if there is a quaternary Hermitian LCD code, then there is a maximal entanglement entanglement-assisted quantum code (see e.g. [11] and [12]). From this point of view, quaternary Hermitian LCD codes play an important role in the study of maximal entanglement entanglement-assisted quantum codes.

It is a fundamental problem to determine the largest minimum weight among all quaternary Hermitian LCD codes for a given pair . It was shown that if and if for  [10] and [12]. In this paper, we give some conditions for the nonexistence of quaternary Hermitian LCD codes with large minimum weights. We give a classification of (unrestricted) quaternary codes for and quaternary codes. Using the above classification and the classification in [3], we completely determine the largest minimum weight among all quaternary Hermitian LCD codes of dimension . In addition, for a positive integer , a maximal entanglement entanglement-assisted quantum codes is constructed for the first time from a quaternary Hermitian LCD code.

This paper is organized as follows. In Section 2, we prepare some definitions, notations and basic results used in this paper. In Section 3, we give characterizations of quaternary Hermitian LCD codes. It is shown that there is no quaternary Hermitian LCD code for and (Theorem 3.3). In addition, if , and , then there is no quaternary Hermitian LCD code with , where denotes the minimum weight of a quaternary code and denotes the Hermitian dual code of . If , and there is no quaternary Hermitian LCD code with , then there is no quaternary Hermitian LCD code with (Theorem 3.4). In Section 4, from the classification of quaternary codes of dimension by Bouyukliev, Grassl and Varbanov [3], we determine for . We emphasize that there is a quaternary Hermitian LCD code. This implies the existence of a quaternary Hermitian LCD code for (Proposition 5.4). We also give a classification of quaternary codes for and quaternary codes. In Section 5, we completely determine as follows: if , if and if (Theorem 5.1). This result is mainly obtained by applying Theorems 3.3 and 3.4 to the classification of some quaternary codes of dimension given in Section 4. In Section 6, as a consequence of Proposition 5.4, we show that there is a maximal entanglement entanglement-assisted quantum code for . This determines the largest minimum weight among maximal entanglement entanglement-assisted quantum codes as . Finally, in Appendix, we give a proof of Proposition 2.4.

2 Preliminaries

In this section, we prepare some definitions, notations and basic results used in this paper.

2.1 Definitions and notations

We denote the finite field of order by , where . For any , the conjugation of is defined as . Throughout this paper, we use the following notations. Let and

denote the zero vector and the all-one vector of length

, respectively. Let

denote the zero matrix of appropriate size. Let

denote the identity matrix of order

and let denote the transpose of a matrix . For a matrix , the conjugate matrix of is defined as . For a matrix , we denote by the matrix

A quaternary code is a -dimensional vector subspace of . A generator matrix of a quaternary code is a matrix such that the rows of the matrix generate . The weight of a vector is the number of non-zero components of . A vector of is called a codeword of . The minimum non-zero weight of all codewords in is called the minimum weight of . A quaternary code is a quaternary code with minimum weight . The weight enumerator of a quaternary code is the polynomial , where denotes the number of codewords of weight in . Two quaternary codes and are equivalent if there is an monomial matrix over with .

For any (unrestricted) quaternary code, the Griesmer bound is given by

(1)

Throughout this paper, we use the following notation:

(2)

where denotes the set of nonnegative integers.

The Euclidean dual code of a quaternary code is defined as where for . The Hermitian dual code of a quaternary code is defined as where for . A quaternary code is called Euclidean linear complementary dual if . A quaternary code is called Hermitian linear complementary dual if . These two families of quaternary codes are collectively called linear complementary dual (LCD for short) codes. Note that quaternary Hermitian LCD codes are also called zero radical codes (see e.g. [11] and [12]).

A quaternary code is called Hermitian self-orthogonal if . It is known that a quaternary code is Hermitian self-orthogonal if and only if is even [13, Theorem 1]. In addition, a quaternary code is Hermitian self-orthogonal if and only if for a generator matrix of .

A - design is a pair of a set of points and a collection of -element subsets of (called blocks) such that every -element subset of is contained in exactly blocks. The number of blocks that contain a given point is traditionally denoted by , and the total number of blocks is . Often a - design is simply called a -design. A -design is called symmetric if . A -design can be represented by its incidence matrix , where if the -th point is contained in the -th block and otherwise.

2.2 Quaternary Hermitian LCD codes

The following characterization gives a criteria for quaternary Hermitian LCD codes and is analogous to [14, Proposition 1].

Proposition 2.1 ([8, Proposition 3.5]).

Let be a quaternary code. Let be a generator matrix of . Then is Hermitian LCD if and only if is nonsingular.

Throughout this paper, we use the above characterization without mentioning this.

Lemma 2.2.

Suppose that there is a quaternary Hermitian LCD code . If , then .

Proof.

Suppose that . Then some column of a generator matrix of is . By deleting the column, a quaternary Hermitian LCD code is constructed. This contradicts the assumption that . ∎

Lemma 2.3.

Let and be generator matrices of a quaternary Hermitian LCD code and a quaternary Hermitian self-orthogonal code, respectively. Then the code with generator matrix is a quaternary Hermitian LCD code with .

Proof.

The straightforward proof is omitted. ∎

2.3 Determination of

Suppose that there is an (unrestricted) quaternary code. By the Griesmer bound (1), we have

(3)

Lu, Li, Guo and Fu [12, Lemma 3.1] constructed a quaternary Hermitian LCD code for and , and a quaternary Hermitian LCD code for and . The following proposition is mentioned in [12], by quoting [10].

Proposition 2.4 (Li [10]).

If , then there is no quaternary Hermitian LCD code.

Remark 2.5.

In Appendix, we give a proof of the above proposition for the sake of completeness.

Hence, one can determine as follows:

for .

3 Nonexistence of some quaternary Hermitian LCD codes

In this section, we give results on the nonexistence of some quaternary Hermitian LCD codes.

An easy counting argument yields the following lemma. We give a proof for the sake of completeness. Recall that and denote the number of blocks of a - design and the number of blocks containing a given point of , respectively.

Lemma 3.1.

Let and be positive integers. Let be a vector of . Suppose that there is a - design . Let be the incidence matrix of . If each entry of the -matrix is at least , then

for any .

Proof.

Fix a point of . Define the following sets:

where . Let denote the -th entry of the -matrix . Then we have

Since , we have

Since and , we have

This completes the proof. ∎

According to [12], we define the matrices by inductive constructions as follows:

The matrix is a generator matrix of the quaternary simplex code. It is known that the quaternary simplex code is a constant weight code. More precisely, the code contains codewords of weights and only. Thus, for , the quaternary simplex code is even. By [13, Theorem 1], the quaternary simplex code is Hermitian self-orthogonal for .

Let be the -th column of . For a vector with , we define a matrix:

For a quaternary code with , there is a vector such that is equivalent to a code with generator matrix . We denote the code by .

Lemma 3.2.

Suppose that , and . If a quaternary code has minimum weight at least , then

(4)

for any .

Proof.

It is well known that the supports of the codewords of weight in the quaternary simplex code form a symmetric - design for (see e.g. [4, p. 8]). As the -design in Lemma 3.1, consider the symmetric - design. Since , it follows from the structure of that each entry of the matrix is at least . The result follows from Lemma 3.1. ∎

If we write , then (see (2) for ).

Theorem 3.3.

Suppose that . If , then there is no quaternary Hermitian LCD code.

Proof.

Write . Suppose that there is a quaternary Hermitian LCD code . Since , by Lemma 2.2. Thus, we may assume that is equivalent to a code for some vector . Consider the conditions given in (4). Since we have

by Lemma 3.2, we have for each . This means that . Since , we have

By [13, Theorem 1], is quaternary Hermitian self-orthogonal, which is a contradiction. ∎

Set

for positive integers and . The following theorem is one of the main results in this paper.

Theorem 3.4.

Suppose that and .

  • Suppose that . Then there is no quaternary Hermitian LCD code with .

  • Suppose that . If there is no quaternary Hermitian LCD code with , then there is no quaternary Hermitian LCD code with .

Proof.

Suppose that there is a quaternary Hermitian LCD code with . Then is equivalent to a code with generator matrix for some vector . Since , by Lemma 3.2, we have

for each . Thus, at least columns of the matrix are , then we obtain a matrix of the following form:

by permuting columns of . Here, is a matrix, where

The code with generator matrix is a quaternary Hermitian self-orthogonal code, where

Since , we have . Since , we have

  • Suppose that . Since is a matrix,

    which is a contradiction.

  • Suppose that . Let be the quaternary code with generator matrix . Since , is a quaternary Hermitian LCD code. It follows from the form of that . Let denote the minimum weight of . By Lemma 2.3, . Since is a constant weight code, there is a codeword of weight in . Thus, then we have

    Therefore, there is a quaternary Hermitian LCD code with .

This completes the proof. ∎

Remark 3.5.

If , then we have

since

4 Quaternary codes of dimension 3

In this section, a classification of (unrestricted) quaternary codes of dimension is done for some lengths by using computer calculations (see Lemma 5.5 for the motivation of our classification). All computer calculations were done by programs in Magma [2] and programs in the language C.

4.1 Classification method

A shortened code of a quaternary code is the set of all codewords in which are in a fixed coordinate with that coordinate deleted. A shortened code of a quaternary code with is a quaternary code if the deleted coordinate is a zero coordinate and a quaternary code with otherwise.

By considering the inverse operation of shortening, every quaternary code with is constructed from some quaternary code with . By considering equivalent quaternary codes, we may assume that a quaternary code has the following generator matrix:

(5)

where and . For the generator matrix (5) of each of all inequivalent quaternary codes with , consider the generator matrices where

(6)

where , under the condition that for and . Here, we consider a natural order on the elements of as follows . In this way, all quaternary codes, which must be checked further for equivalences, are constructed. By checking equivalences among these codes, we complete a classification of quaternary codes.

4.2 Lengths up to 35

Here we investigate the values for lengths . Let denote the largest minimum weight among all (unrestricted) quaternary codes (see [7] for the current information on ). From [12, Tables 3 and 4], we know for

For , Bouyukliev, Grassl and Varbanov [3] completed the classification of (unrestricted) quaternary codes. The number of the inequivalent quaternary codes are given in [3, Table 3]. Based on the number given in [3, Table 3], we reconstructed all inequivalent quaternary codes for

by using the method in Section 4.1. Then we found that for the above lengths except . For length , we found that . For the remaining lengths, from [12, Tables 3 and 4], we know . This determines the largest minimum weight for lengths , where the results are listed in Table 1. In the table, the reference about the existence of quaternary Hermitian LCD codes is also listed.

 

Reference Reference
4 1  [12, Table 3] 20 14  [12, Table 3]
5 2  [12, Table 3] 21 15  [12, Table 3]
6 3  [12, Table 3] 22 15  [12, Table 4]
7 4  [12, Table 3] 23 16  [12, Table 4]
8 5  [12, Table 3] 24 17  [12, Table 4]
9 6  [12, Table 3] 25 18  [12, Table 4]
10 6  [12, Table 3] 26 19
11 7  [12, Table 3] 27 19  [12, Table 4]
12 8  [12, Table 3] 28 20  [12, Table 4]
13 9  [12, Table 3] 29 21  [12, Table 4]
14 9  [12, Table 3] 30 22  [12, Table 4]
15 10  [12, Table 3] 31 22  [12, Table 4]
16 11  [12, Table 3] 32 23  [12, Table 4]
17 12  [12, Table 3] 33 24  [12, Table 4]
18 13  [12, Table 3] 34 25  [12, Table 4]
19 13  [12, Table 3] 35 25  [12, Table 4]

 

Table 1:

We give details for the case . There are five inequivalent quaternary codes [3, Table 3]. We verified that one of them is Hermitian LCD. This code has the following generator matrix:

and the following weight enumerator:

4.3 Lengths 36, 40, 43, 48, 52, 56 and 64

By using the method in Section 4.1, a classification of (unrestricted) quaternary codes for and quaternary codes was done. These codes have minimum weights . To save space, the results are given only.

Proposition 4.1.
  • There are two inequivalent quaternary codes , none of which is Hermitian LCD for .

  • There are ten inequivalent quaternary codes , none of which is Hermitian LCD.

  • There are five inequivalent quaternary codes , none of which is Hermitian LCD for .

  • There are six inequivalent quaternary codes , none of which is Hermitian LCD.

  • There are inequivalent quaternary codes , none of which is Hermitian LCD.