A Note on Self-Dual Generalized Reed-Solomon Codes

05/24/2020 ∙ by Weijun Fang, et al. ∙ Nankai University Tsinghua University Capital Normal University NetEase, Inc 0

A linear code is called an MDS self-dual code if it is both an MDS code and a self-dual code with respect to the Euclidean inner product. The parameters of such codes are completely determined by the code length. In this paper, we consider new constructions of MDS self-dual codes via generalized Reed-Solomon (GRS) codes and their extended codes. The critical idea of our constructions is to choose suitable evaluation points such that the corresponding (extended) GRS codes are self-dual. The evaluation set of our constructions is consists of a subgroup of finite fields and its cosets in a bigger subgroup. Four new families of MDS self-dual codes are obtained and they have better parameters than previous results in certain region. Moreover, by the Mobius action over finite fields, we give a systematic way to construct self-dual GRS codes with different evaluation points provided any known self-dual GRS codes. Specially, we prove that all the self-dual extended GRS codes over F_q with length n< q+1 can be constructed from GRS codes with the same parameters.

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1 Introduction

An -linear code over is just a subspace of with dimension and minimum Hamming distance , where is given as

The code rate defined by and the minimum Hamming distance are two important metrics of the code. One of the trade-off between these parameters is the well-known Singleton bound:

An -linear code achieving the Singleton bound is called a maximum distance separable (MDS) code. MDS codes have various interesting properties and wide applications both in theoretical and practice. On the one hand, MDS codes are closely related to many other mathematical aspects, such as orthogonal arrays in combinatorial design and -arcs in finite geometry (cf. [1, Chap. 11]). On the other hand, MDS codes also have widespread applications in data storage, such as CD ROMs and coding for distributed storage systems (see [2]). In particular, the most commonly used class of MDS codes is the well-known Reed-Solomon codes, which not only have nice theoretic properties, but also have been extensively applied in engineering due to their easy encoding and efficient decoding algorithm.

The Euclidean inner product in is given as

where and . For any -linear code , we define the dual code of as

One of important connections between and its dual is the well-known MacWilliams identity (cf.[1]). We say is a self-dual code if . Self-dual codes is a very interesting and important class of linear codes. The length of a self-dual code is obviously even and the dimension is equal to half of length. The existence result of self-dual codes was given by Pless in [3], which showed that a -ary self-dual code of even length exists if and only if is a square element in . Self-dual codes have also some applications in other aspects, such as linear secret sharing schemes (see [4, 5]) and unimodular integer lattices (see [6, 7]). In past decades, it attracts lots of attentions for investigating the codes which are both MDS and self-dual. Such codes are called MDS self-dual codes. Note that the dimension and minimum distance of an MDS self-dual code over are equal to and , respectively. Therefore, a natural question is that for which even an MDS self-dual code over of length exists. According to the well-known MDS conjecture, it is expected to determine the existence of -ary MDS self-dual codes of length for all possible .

Recent years, several progress have been made in this topic (see [8, 9, 10, 11, 12, 13, 14]). For the case where is even, it was shown by Grassl and Gulliver in [15] that there is an MDS self-dual code of even length for all . By using the properties of cyclic and constacyclic codes, some new MDS self-dual codes were obtained in [16] and [17]. A systematic construction of MDS self-dual codes was first proposed by Jin and Xing in [18] via generalized Reed-Solomon (GRS) codes over finite fields. A subset of elements of , called evaluation set, with some special properties can be used to construct desired GRS codes which are self-dual. Since then, GRS codes becomes one of the most commonly used tools to construct MDS self-dual codes. In [19] and [20], the authors generalized this method to extended GRS codes with general length. The evaluation set with special structures, such as a multiplicative subgroup of , a subspace of and their cosets, are considered for constructing self-dual GRS codes. Zhang and Feng [21, 22] presented a unified approach to MDS self-dual codes and obtain some new codes via cyclotomy. Fang et al. [23, 24, 25] constructed new families of self-dual GRS codes via two disjoint multiplicative subgroups of and their cosets. In [26], by using the technique of algebraic geometry codes, Sok constructed several new families of MDS self-dual codes. In the following Table 1, we summary some known results about the construction of MDS self-dual codes.

References
even [15]
odd [15, 18]
[18]
,
[18]
[18]
and [17]
[17]
[20]
[20]
,
odd, even
, ,
and
[20]
,
odd, is even
, ,
and
[20]
odd , [20, 19]
[19]
,
odd,
, even and [19]
,
odd,
, even,
and
[19]
,
odd,
, odd,
and
[19]
,
odd,
, odd,
and
[19]
, odd
, even,
and
[23]
, odd
, odd, ,
and
[23]
, odd
, , even
(except are even and ),
and
[23]
, odd
,
even, even, , and even
[23]
, odd
,
even, even, , and even
[23]
, odd
, even,
and
[24]
, odd
, odd, ,
and
[24]
, odd
, even, ,
and
[24]
,
, even,
[25]
,
, odd,
[25]
Table 1: Some known results on MDS self-dual codes of even length

In this paper, we investigate the construction of MDS self-dual codes with new parameters by using (extended) GRS codes over finite fields. The main idea of our constructions is to consider two multiplicative subgroups and of , where is a subgroup of . We consider the union of some costes of in as the evaluation set. Then we present two new constructions (see Theorems 1 and 2) of MDS self-dual codes via Lemmas 2 and 3. Moreover, by considering some automorphisms of GRS codes, we give a systematic way to construct self-dual GRS codes provided any known self-dual GRS code. As a corollary, all the self-dual extended GRS codes over finite field with length can be constructed from self-dual GRS codes with the same parameters.

We list the parameters of our new MDS self-dual codes as follows. Specifically, if one of the following conditions holds, then there exists a self-dual GRS code of length over , where and is a power of an odd prime .

(i)

is even, , is even, and ; (see Theorem 1 (i))

(ii)

is even, , , and ; (see Theorem 1 (iii))

(iii)

is even, , both and are even, where , , and ; (see Theorem 2 (i))

(iv)

is even, , both and are even, or is odd and is even with ; (see Theorem 2 (ii))

The rest of this paper is organized as follows. In Section 2, we briefly introduce some basic notations and results about generalized Reed-Solomon codes and MDS self-dual codes. In Section 3, four new families of self-dual GRS codes are constructed. In Section 4, we give a systematic way to construct self-dual GRS codes from known self-dual GRS code. Finally, we conclude this paper in Section 5.

2 Preliminaries

In this section, we recall some basic notations and results about generalized Reed-Solomon codes and MDS self-dual codes.

Let be a prime power and be the finite field with elements. Let and be two integers such that and Write the finite field and let be the infinity point. For any , let

be a column vector of

defined as

Definition 1.

Let be a subset of and be a vector in . For any if , the linear code with generator matrix is called the generalized Reed-Solomon code. If , then the linear code is called extended generalized Reed-Solomon code. We call them (extended) generalized Reed-Solomon (GRS) code with evaluation set and scaling vector , and denoted by .

It is well-known that is an -MDS code and its dual code is also a GRS code.

Recently, lots of research work on construction of MDS self-dual codes have been done by using GRS codes. The key point of these constructions is to choose suitable evaluation set such that the corresponding GRS code is self-dual for some scaling vector . In the following, we introduce some related notations and results.

Let be a subset of , we define the polynomial over as

For any element , we define

The properties of and are given as follows, which were first obtained in [21]. We present its proof for the completeness.

Lemma 1.

[21, Lemma 3.1]

(i)

Let be a subset of , then for any

where is the derivative of .

(ii)

Let be pairwise disjoint subsets of , and . Then for any ,

Proof.
(i)

The conclusion follows from

(ii)

Note that . Thus

Part (ii) then follows from Part (i) and the fact that for any .

Let be the quadratic character of , that is if is a square in and if is a non-square in . The following Lemmas 2 and 3 are the key lemmas for the construction of MDS self-dual codes, which have been used in the literature with some equivalent forms. The reader may refer to [18, 19, 20, 21, 23, 24, 25] for more details on their proofs.

Lemma 2.

Suppose is even. Let be a subset of of size , such that for all , are the same. Then there exists a vector such that is self-dual. Consequently, there exists a -ary MDS self-dual code of length .

Lemma 3.

Suppose is odd. Let be a subset of of size , such that for all , . Then there exists a vector such that is self-dual. Consequently, there exists a -ary MDS self-dual code of length .

3 New Constructions of MDS Self-Dual Codes

In this section, based on Lemmas 2 and 3, we give two new constructions of MDS self-dual codes via different multiplicative subgroups of finite fields. Throughout this section, we suppose that and , where is an odd prime.

Let be a positive integer with . We write , where and . Then . Note that , hence

Let be a primitive element of . Denote

Then . Note that , thus , which deduces that is a subgroup of . Then there exist such that represent all cosets of .

Let and . Denote ,

(1)

and

(2)

Based on (1) and (2), we give our first construction as follows.

Theorem 1.

Let . Suppose and , where and . Let and ,

(i)

if both and are even, then there exists a -ary MDS self-dual code of length ;

(ii)

if is odd, then there exists a -ary MDS self-dual code of length ;

(iii)

if is even, then there exists a -ary MDS self-dual code of length .

Proof.

(i): Let be defined as (1). Then . For any ,

Given , suppose for some and . Then by Lemma 1,

Since , for some . Thus is a square element of . Since is even, thus is a square. Note that for any ,

Hence , which is a square element of . Thus is a square element of , i.e., for all . The Part (i) then follows from Lemma 2. (ii) and (iii): Let be defined as (2). Then . Similar to the proof of Part (i), for any , we have

Moreover,

Thus for any , , the conclusions of Part (ii) and Part (iii) then follow from Lemma 2 and Lemma 3, respectively. ∎

Remark 1.

In [24, Theorems 1, 2 and 3], the authors constructed MDS self-dual codes of length , and , respectively, where and . When is odd, then since . Thus . The result of our Theorem 1 (ii) is equivalent to [24, Theorem 2]. When both and are even, then it can be verified that . Thus , i.e., . Thus in this case, our Theorem 1 (i) and (iii) are better than [24, Theorems 1 and 3], respectively.

Example 1.

In Theorem 1 (i) and (iii), let , , . Then and . Let , then . So we can obtain two MDS self-dual codes of length 156 and 158 over from Theorem 1 (i) and (iii), respectively. Compared with the parameters of MDS self-dual codes given in Table 1, these two codes are new.

In the following, we consider another multiplicative subgroup of . Suppose . We write , where and . Then . Note that , hence

Let be a primitive element of . Denote

Then . Note that , thus , which deduces that is a subgroup of . Then there exist such that represent all cosets of .

Let and . Denote ,

(3)

and

(4)

Based on (3) and (4), we give our second construction of MDS self-dual codes as follows.

Theorem 2.

Let . Suppose and , where and . Let and ,

(i)

if both and are even, then there exists a -ary MDS self-dual code of length ;

(ii)

if both and are even, or is odd and is even with , then there exists a -ary MDS self-dual code of length .

Proof.

(i): Let be defined as (3). Then . For any ,

Given , suppose for some and . Then by Lemma 1,

For any , since , we write for some integer . Thus

Hence

Denote . Then

Hence

Note that . Thus there exists an integer such that

Note that which is a square since both and are even. Thus for all . The Part (i) then follows from Lemma 2. (ii): Let be defined as (4). Then . According to the proof of Part (i), for any , where and , we have

for some integer . If both and are even, then is a square in , i.e., . If is odd and is even, then