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: , the code 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 the 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], combinatorics [References, References] and other related areas.
MDS selfdual codes have good properties due to its optimality with respect to the Singleton bound and its selfduality, which have attracted a lot of attentions in recent years. Some people constructed MDS selfdual codes through the way of orthogonal designs, see [References, References, References, References]. Some people utilized constacyclic codes to construct MDS selfdual codes, see [References, References, References, References]. Some people make the construction by ReedSolomon codes, generalized ReedSolomon codes or extended generalized ReedSolomon codes, see [References, References, References, References, References].
Parameters of MDS selfdual codes are completely determined 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 on the finite fields of odd characteristic is still far from complete. For example, if , more than 3000 MDS selfdual codes with different even lengths possibly exist assuming MDS conjecture is valid. But up to now, only 702 MDS selfdual codes of different even lengths are able to be 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 construction of MDS selfdual codes are depicted in Table 1. Some other constructions, using building up technique can be found in [References]. Based on [References], [References] and [References], in this paper, we give more constructions on MDS selfdual code. 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 work.
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  
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] 
2 Preliminaries
In this section, we introduce some basic notations 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 any 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 below:
(1) 
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 with such that the code defined in (1) is an MDS selfdual code of length .
Moreover, extended code can also be applied into the construction of MDS selfdual codes. We can choose the two tuples and in the same way as code. Then the extended code of length associated with and is defined as follows:
(2) 
where is the coefficient of in .
It is also wellknown that is a ary MDS code and its dual is also an MDS code [References, Chapter 11].
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 , we have
3 Main Results
In this section, we will give several new constructions of MDS selfdual codes utilizing the additive group structure on and the multiplicative group structure of .
Theorem 1.
Let , where is an odd prime power. Suppose . For , assume is even.
(i). If is even and , there exists a ary MDS selfdual code.
(ii). Assume . There exists a ary MDS selfdual code except 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). We choose distinct elements such that . Denote and . Let be a set of coset representatives of and
Then 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 . Then
So .
Let be a generator of such that and . So
It follows that
Note that . We take . Since is even, we obtain that . Choose with . Define
Then by Lemma 2.1, is an MDS selfdual code.
Therefore, we know that there exists a ary MDS selfdual code with the length of .
(ii). Similarly as (i), we let
We will find such that is an MDS selfdual code of length .
For any and for any , , we have
and
Denote . We can obtain that for some , in the same way as (i). We discuss in the following.
: If is odd and is even, we have is even. It follows that .
: If is even and , we can choose such that is an even integer. 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 . So in all three cases, one has .
It is easy to verify that . We choose and , with . Define
Then by Lemma 2.2, is an MDS selfdual code with the length of , except that is even, is even and .
∎
Example 3.1.
Let , , and . Then . It is easy to verify that is even. By Theorem 1 (i), we know there exists the MDS selfdual code of length . This is a new parameter of MDS selfdual code which was not reported in any previous work.
Theorem 2.
Let , where is an odd prime power. Suppose . For , we assume is odd and . Then there exists a ary MDS selfdual code over .
Proof.
The elements and are the same as in Theorem 1 (i). We choose distinct even elements , where . We take and . It is easy to see that is even. Let
The proof is similar as in Theorem 1 (i). We deduce that
Let . We can obtain in the same way as Theorem 1 (i). Since is odd, and are even, then is even. It follows that .
Since is odd, then is even, that is . Note that . Therefore, we have . Choose , with . Define
By Lemma 2.2, is an MDS selfdual code with the length of . ∎
Example 3.2.
Let , , and . Then . It is easy to verify that . By Theorem 2, we know there exists 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 , be even, and . Assume that .
(i). Assume . If and are even, there exists a ary MDS selfdual code.
(ii). Assume . There exists a ary MDS selfdual code.
Proof.
We choose the elements and in the same way in Theorem 1 (i). Here is a primitive th root of unity and is a primitive th root of unity in . Let . From the second fundamental theorem of group homomorphism, we get
(i). We choose distinct elements such that and denote .
Let be a set of coset representatives of and
Then the entries of are distinct elements of . We will show that there exists such that is an MDS selfdual code of length .
Similarly as Theorem 1 (i), we get
Since the order of is , then is a primitive th root of unity. So .
Let . Since , then
which implies .
If both and are even, one has .
Now we obtain . Hence .
Therefore, we know that there exists a ary MDS selfdual code with the length of .
(ii). Similarly as (i), we let
We will find such that is an MDS selfdual code of length .
For any and for any , one has
and
Since the order of is , then , where is a primitive th root of unity. So with . It implies that . Therefore, we know that . Since , then . We choose and , with . Define
Then by Lemma 2.2, is an MDS selfdual code with the length of .
∎
Example 3.3.
Let , , and . Then both and are even. Note that , by Theorem 3 (i), we know there exists 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. Then there exists a ary MDS selfdual code of length of , where .
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