1 Introduction
Let be the finite field with elements, where is a prime power. A linear code of length , dimension and minimum distance over is usually called a ary code. If the parameters of the code attach the Singleton bound: , then is called a maximum distance separable (MDS) code. MDS codes are widely applied in various occasions due to their nice properties, see [References, References, References].
The dual code of a linear code in , denoted by , is a linear subspace of , which is orthogonal to . If , is called a selfdual code. Selfdual codes have important applications in coding theory [References], cryptograph [References, References, References], combinatorics [References, References] and other related areas.
MDS selfdual codes have good properties due to its optimality with respect to the Singleton bound and their selfduality, which have attracted a lot of attention in recent years. There are various ways to construct MDS selfdual codes. They mainly are: (1). orthogonal designs, see [References, References, References]; (2). building up technique, see [References, References]; (3). constacyclic codes, see [References, References, References]; (4). (generalized and/or extended) ReedSolomon codes, see [References. References, References, References, References, References, References].
Parameters of MDS selfdual codes are completely characterized by their lengths , that is, . Therefore, the problem for constructing different MDS selfdual codes can be transformed to find MDS selfdual codes with different lengths. In [References], Grassl and Gulliver showed that the problem has been completely solved over the finite fields of characteristic . But the constructions of MDS selfdual codes on finite fields of odd characteristic are still far from complete. For example, if , more than 3000 MDS selfdual codes with different even lengths possibly exist assuming MDS conjecture is valid (MDS conjecture says that the length of nontrivial ary MDS code with odd prime power, is bounded by ). But up to now, only 702 ary MDS selfdual codes of different even lengths have been constructed. In [References], Jin and Xing constructed some classes of new MDS selfdual codes through generalized ReedSolomon codes. In [References], Yan generalized the technique in [References] and constructed several classes of MDS selfdual codes via generalized ReedSolomon codes and extended generalized ReedSolomon codes. In [References], Labad, Liu and Luo produced more classes of MDS selfdual codes based on [References] and [References]. All the known results on the systematic constructions of MDS selfdual codes are depicted in Table 1.
even  Reference  
even  [References]  
odd  [References]  
odd  ,  [References] 
odd  ,  [References] 
, prime and odd  [References]  
, , odd  , odd and prime  [References] 
, odd,  , even and  [References] 
, odd, 
, even , and  [References] 
, odd, 
, odd , and  [References] 
, odd,  , odd , and  [References] 

[References]  
for any  [References]  
, odd 
, even and  [References] 
, odd 
, odd and  [References] 

[References]  
[References]  
, odd prime 
,  [References] 
, odd prime  , ,  [References] 
, odd  , , even  [References] 
, odd  , odd, and  [References] 
, odd  , even, and  [References] 
, odd prime  , and , even  [References] 
Based on [References], [References] and [References], we give more constructions of MDS selfdual codes in this paper. Among our constructions, there are several MDS selfdual codes with new parameters (see Table 2). In particular, for square , we can produce much more MDS selfdual codes than previous works.
This paper is organized as follows. In Section 2, we will introduce some basic knowledge and useful results on (extended) generalized ReedSolomon codes. In Section 3, we will present our main results on the constructions of MDS selfdual codes. In Section 4, we will make a conclusion.
even  Reference  
, odd  , , even  Theorem 1 (i)  
, odd 

Theorem 1 (ii)  
, odd  , odd, and  Theorem 2  
, odd 

Theorem 3 (i)  
, odd 

Theorem 3 (ii)  
, odd prime  ,  Theorem 4  
, odd prime  , and , even  Theorem 5 
2 Preliminaries
In this section, we introduce some basic notation and useful results on (extended) generalized ReedSolomon codes (or (extended) codes for short). Readers are referred to [References, Chapter 10] for more details.
Let be the finite field with elements and be an integer with . Choose two tuples and , where , ( may not be distinct) and , are distinct elements in . For an integer with , the code of length associated with and is defined as follows:
(1) 
The code is a ary MDS code and its dual is also MDS [References, Chapter 11].
We define
Let denote the set of nonzero squares of . The following result is useful in our constructions and it has been shown in [References].
Lemma 2.1.
([References], Corollary 2.4) Let be an even integer and . If there exists such that for all , then there exists with such that the code defined in (1) is an MDS selfdual code of length .
Moreover, extended code can also be applied to the construction of MDS selfdual codes. For and , the extended code of length associated with and is defined as follows:
(2) 
where is the coefficient of in . The code is a ary MDS code and its dual is also MDS [References, Chapter 11].
We present another two useful results, which have been shown in [References].
Lemma 2.2.
([References], Lemma 2) Let be an even integer and . If for all , then there exists with such that the code defined in (2) is an MDS selfdual code of length .
Lemma 2.3.
([References], Lemma 3) Let be a positive integer and let be a primitive th root of unity. Then for any ,
3 Main Results
In this section, we will give several new constructions of MDS selfdual codes utilizing the multiplicative group structure of and the additive group structure on .
Theorem 1.
Let , where is an odd prime power. Suppose . For , and even,
(i). if is even and , then there exists a ary MDS selfdual code.
(ii). if , then there exists a ary MDS selfdual code except the case that is even, is even and .
Proof.
Let be a primitive th root of unity in and be the cyclic group of order . By the second fundamental theorem of group homomorphism, we have
(i). Let be a set of coset representatives of with . Denote by , and
Obviously, the entries of are distinct in . We will show that there exists such that is an MDS selfdual code of length .
Note that . By Lemma 2.3, for any and , we deduce
Let . We calculate
So . Let be a generator of such that and . Then
It follows that
Note that . We take . Since is even, we obtain that . Choose with . Define
By Lemma 2.1, is an MDS selfdual code.
Therefore, there exists a ary MDS selfdual code with length .
(ii). As in (i), we let
We will find such that is an MDS selfdual code of length .
For any and for any , ,
and
Denote . We obtain for some , in the same way as (i). The following cases are considered.
: If is odd and is even, we have is even. It follows that .
: If is even and , we can choose such that is even. It follows that is even. Hence .
: If is even, is odd and , we can choose such that is an odd integer. It follows that is even. Hence .
Note that . As a result, one always has .
It is easy to verify that . We choose and , with . Define
By Lemma 2.2, is an MDS selfdual code with length , except the case that is even, is even and .
∎
Example 3.1.
Let , , and . Then . By Theorem 1, there exists MDS selfdual code of length . This is a new parameter of MDS selfdual code.
Theorem 2.
Let , where is an odd prime power. Suppose . If , is odd and , then there exists a ary MDS selfdual code over .
Proof.
Recall and in the proof of Theorem 1 (i). Choose with and () even. Denote by distinct and
The main goal is to find such that is an MDS selfdual code. Similarly as in Theorem 1 (i), for , and , we deduce that
Let . We can obtain in the same way as Theorem 1 (i). Since is odd, and are even, it follows that is even which implies .
Since is odd, it implies that . Note that . Therefore, . Choose , with . Define
By Lemma 2.2, is an MDS selfdual code with length . ∎
Example 3.2.
If , , and , then . By Theorem 2, there exists an MDS selfdual code of length . This is a new parameter of MDS selfdual code which has not been covered by previous work.
Theorem 3.
Let , where is an odd prime power. Let , even, and . For ,
(i). if , both and are even, then there exists a ary MDS selfdual code.
(ii). if , then there exists a ary MDS selfdual code.
Proof.
Let be a primitive th root of unity and be a primitive th root of unity in . Let . From the second fundamental theorem of group homomorphism,
(i). We choose distinct elements such that and denote by . Let be a set of coset representatives of and
Obviously, the entries of are distinct in . We will show that there exists such that is an MDS selfdual code of length .
Similarly as Theorem 1 (i),
Note that the order of is . Then is a primitive th root of unity and . Let . Since , it follows that
which implies . If both and are even, then . Now we obtain . Hence . Choose with . Define
According to Lemma 2.1, is an MDS selfdual code with length .
(ii). As in (i), we let
We will find such that is an MDS selfdual code of length .
For any and for any , one has
and
The order of is , which implies that since . Therefore, . Since , . We choose and , with . Define
According to Lemma 2.2, is an MDS selfdual code with length .
∎
Example 3.3.
If , , , and , then both and are even. Note that . By Theorem 3, there exists a ary MDS selfdual code of length . This MDS selfdual code has not been reported in any previous references.
Theorem 4.
Let , where is an odd prime and is a positive integer. There exists a ary MDS selfdual code of length , where .
Proof.
Denote by . Let be an dimensional
vector subspace of
, with . Choose , such that . Let , and . A routine calculation shows that
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