1 Introduction
Let be a prime power and be a finite field of cardinality . A linear code over , denote by is a linear subspace of with dimension and minimal (Hamming) distance . If the parameters can reach the Singleton bound, that is , then we call be a maximum distance separable (MDS) code. Denote the (Euclidean) dual code of by . If , then is called a selfdual code.
Both MDS codes and selfdual codes have theoretical interest and practical importance. Thus the study of MDS selfdual codes have attracted a lot of interest, especially for the construction of MDS selfdual codes. MDS selfdual codes can be constructed in various ways, which 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].
It is noted that the parameters of MDS selfdual codes can be completely determined by the code length . For the finite field of even characteristic, Grassl and Gulliver completely determined the MDS selfdual codes of length for all in [References]. In [References, References, References], the authors obtained some new MDS selfdual codes through cyclic, constantcyclic and negacyclic codes. In [References], Jin and Xing firstly presented a general and efficient method to construct MDS selfdual codes by utilizing generalized ReedSolomn (GRS) codes. Afterwards, many researchers start to construct MDS selfdual codes via GRS codes. In [References], Zhang and Feng gave a nonexistence result, that is, when and , we can’t construct MDS selfdual code of length . We list all the known results on the systematic constructions of MDS selfdual codes, which 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  , , even  [References]  
, odd 

[References]  
, odd  , odd, and  [References]  
, odd 

[References]  
, odd 

[References]  
, even, odd prime  with , , and  [References]  
, even, odd prime 

[References]  
with  [References] 
In odd characteristic case, the length of MDS selfdual codes is restricted among the above known results. For instance, in Row 7 of Table 1 (see [References]), the length must satisfy the restrictive conditions “ is even and ”. So it can produce a few MDS selfdual codes with different lengths. In this paper, we produce approximately many ary MDS selfdual codes with different lengths.
2 Preliminaries
In this section, we introduce some basic notations and useful results on selfdual codes and GRS codes.
Let be a linear code over . The (Euclidean) dual of is defined as follows:
If , then is called a selfdual code.
For , we choose two tuples and , where , ( may not be distinct) and , are distinct elements in . Then the GRS code of length associated with and is defined below:
(1) 
where .
It is wellknown that the code is a ary MDS code and its dual code is also an MDS code [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 a ary MDS selfdual code of length .
The next lemma has been shown in [References] and is very useful for the proof of our main results.
Lemma 2.2.
([References], Lemma 3) Let be a positive integer and let be a primitive th root of unity. Then for any , we have
3 Main Results
In this section, we give our main results.
Theorem 1.
Let , where is an odd prime power and . For any , is even and , assume . There exists a ary MDS selfdual code.
Proof.
Let be a primitive element of , , and . Denote by and and choose
Since , it follows that and . Therefore, for all and . By Lemma 2.2, it is necessary to consider two cases.

For and ,
(2) Note that and . We only need to consider . Then
So
that is
(3) with some integer .

For and ,
(4) We consider . Then
So
that is , for some integer .
Remark 3.1.
In Theorem 1, the number of is and the number of is . Thus the number of MDS selfdual codes is .
Example 3.1.
For , we can construct different for which MDS selfdual code of length by utilizing all the previous results (in Table 1). But we can produce different classes of MDS selfdual codes of different lengths by Theorem 1 in this paper. So the results in this paper can construct much more MDS seldual codes than previous works for large square .
Theorem 2.
Let , where is an odd prime power and . For any , and is odd, assume . There exists a ary MDS selfdual code.
Proof.
Let be a primitive element of , , and . Denote by and and choose
Since , it follows that and . Therefore, for all and . By Lemma 2.2, it is necessary to consider two cases.

For and ,
(5) Note that and . We only need to consider . Let . Then . Therefore,
So
that is,
(6) for some integer .

For and ,
(7)
Let and . Then Therefore,
It follows that
that is with some integer . Since and is odd, then it is easy to verify that is even, which yields .
Remark 3.2.
In Theorem 2, the number of is and the number of is . Thus the number of MDS selfdual codes is .
Example 3.2.
For , we can construct different for which MDS selfdual code of length by utilizing all the previous results (in Table 1). But we can produce about different classes of MDS selfdual codes of different lengths by Theorem 2 in this paper. So the results in this paper can construct much more MDS seldual codes than previous works for large square .
4 Conclusion
Utilizing GRS codes, we construct two new classes of MDS selfdual codes. For any large square , we can produce about new MDS selfdual codes with different lengths. The number is already very large, but there is still a long way to go to reach the level of . So there are still a lot of work waiting to do.
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