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 tradeoff between these parameters is the wellknown 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 wellknown ReedSolomon 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 wellknown MacWilliams identity (cf.[1]). We say is a selfdual code if . Selfdual codes is a very interesting and important class of linear codes. The length of a selfdual code is obviously even and the dimension is equal to half of length. The existence result of selfdual codes was given by Pless in [3], which showed that a ary selfdual code of even length exists if and only if is a square element in . Selfdual 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 selfdual. Such codes are called MDS selfdual codes. Note that the dimension and minimum distance of an MDS selfdual code over are equal to and , respectively. Therefore, a natural question is that for which even an MDS selfdual code over of length exists. According to the wellknown MDS conjecture, it is expected to determine the existence of ary MDS selfdual 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 selfdual code of even length for all . By using the properties of cyclic and constacyclic codes, some new MDS selfdual codes were obtained in [16] and [17]. A systematic construction of MDS selfdual codes was first proposed by Jin and Xing in [18] via generalized ReedSolomon (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 selfdual. Since then, GRS codes becomes one of the most commonly used tools to construct MDS selfdual 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 selfdual GRS codes. Zhang and Feng [21, 22] presented a unified approach to MDS selfdual codes and obtain some new codes via cyclotomy. Fang et al. [23, 24, 25] constructed new families of selfdual 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 selfdual codes. In the following Table 1, we summary some known results about the construction of MDS selfdual codes.
References  
even  [15]  
odd  [15, 18]  
[18]  

[18]  
[18]  
and  [17]  
[17]  
[20]  
[20]  


[20]  


[20]  
odd  ,  [20, 19]  
[19]  

, even and  [19]  


[19]  


[19]  


[19]  
, odd 

[23]  
, odd 

[23]  
, odd 

[23]  
, odd 

[23]  
, odd 

[23]  
, odd 

[24]  
, odd 

[24]  
, odd 

[24]  


[25]  


[25] 
In this paper, we investigate the construction of MDS selfdual 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 selfdual codes via Lemmas 2 and 3. Moreover, by considering some automorphisms of GRS codes, we give a systematic way to construct selfdual GRS codes provided any known selfdual GRS code. As a corollary, all the selfdual extended GRS codes over finite field with length can be constructed from selfdual GRS codes with the same parameters.
We list the parameters of our new MDS selfdual codes as follows. Specifically, if one of the following conditions holds, then there exists a selfdual GRS code of length over , where and is a power of an odd prime .
The rest of this paper is organized as follows. In Section 2, we briefly introduce some basic notations and results about generalized ReedSolomon codes and MDS selfdual codes. In Section 3, four new families of selfdual GRS codes are constructed. In Section 4, we give a systematic way to construct selfdual GRS codes from known selfdual GRS code. Finally, we conclude this paper in Section 5.
2 Preliminaries
In this section, we recall some basic notations and results about generalized ReedSolomon codes and MDS selfdual 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 asDefinition 1.
Let be a subset of and be a vector in . For any if , the linear code with generator matrix is called the generalized ReedSolomon code. If , then the linear code is called extended generalized ReedSolomon code. We call them (extended) generalized ReedSolomon (GRS) code with evaluation set and scaling vector , and denoted by .
It is wellknown that is an MDS code and its dual code is also a GRS code.
Recently, lots of research work on construction of MDS selfdual 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 selfdual 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 nonsquare in . The following Lemmas 2 and 3 are the key lemmas for the construction of MDS selfdual 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 selfdual. Consequently, there exists a ary MDS selfdual 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 selfdual. Consequently, there exists a ary MDS selfdual code of length .
3 New Constructions of MDS SelfDual Codes
In this section, based on Lemmas 2 and 3, we give two new constructions of MDS selfdual 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) 
Theorem 1.
Let . Suppose and , where and . Let and ,
 (i)

if both and are even, then there exists a ary MDS selfdual code of length ;
 (ii)

if is odd, then there exists a ary MDS selfdual code of length ;
 (iii)

if is even, then there exists a ary MDS selfdual 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 selfdual 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 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 selfdual codes as follows.
Theorem 2.
Let . Suppose and , where and . Let and ,
 (i)

if both and are even, then there exists a ary MDS selfdual code of length ;
 (ii)

if both and are even, or is odd and is even with , then there exists a ary MDS selfdual 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
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