1 Introduction
A linear code over a finite field of length , dimension and minimum distance is called MDS (maximum distance separable) if it attains the Singleton bound: . MDS codes have been of much interest from many researchers due to their theoretical significant and practical implications, see [References], [References], [References].
For a linear code , we denote the dual of under Euclidean inner product by . The linear code is called selfdual if . Selfdual codes have attracted attention from coding theory, cryptograph and other fields. It has been found various applications in cryptography, in particular secret sharing scheme [References], [References], [References] and combinatorics [References].
More recently, the application of MDS codes renewed the interest in the construction of MDS selfdual codes, see [References], [References], [References], [References]. K. Guenda [References] constructed MDS Euclidean and Hermitian selfdual codes which are extended cyclic duadic codes or negacyclic codes. She also constructed Euclidean selfdual codes which are extended negacyclic codes.
Generalized ReedSolomon () codes is a class of MDS code which has nice algebraic structure. It can be systematically constructed and has been found wide applications in practice. MDS selfdual codes through codes have been studied by L. Jin and C. Xing [References], where they constructed several classes of MDS selfdual codes through codes by choosing suitable parameters. In [References], H. Yan generalizes the technique in [References] and construct several classes of MDS selfdual codes via codes and extended codes.
Since MDS selfdual codes over finite field of even characteristic with any possible parameter have been found in [References]. In this paper, we obtain some new results on the existence of MDS selfdual codes through (extended) codes over finite fields of odd characteristic. Some results in this paper extend those of [References] and [References]. Comparing to previous works, for some fixed square prime power , our construction will produce more ary MDS selfdual codes.
This paper is organized as follows. In Section 2 we will introduce some basis knowledge and auxiliary results on codes and extended codes. In particular, Corollary 2.1 and Corollary 2.2 give a criterion for an (extended) code to be selfdual. In Section 3 we will present our main results on the construction of MDS selfdual codes. Our main tools are Corollary 2.1 and Corollary 2.2. We choose suitable parameters to make the conditions in Corollary 2.1 and Corollary 2.2 hold.
^{0}^{0}footnotetext: Some results in Table 1 are overlapped. For instance, columns 6 and 7 in [References] are special cases of column 4 in [References].  
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] 
2 Generalized ReedSolomon codes
In this section, we introduce some basic notations and results on generalized ReedSolomon code. Throughout this paper, let be a finite field with elements, and let be a positive integer with . Choose to be an tuple of distinct elements of . Put with . For an integer with , then linear code
(1) 
is the generalized ReedSolomon or code.
The code has a generator matrix
It is wellknown that the code is a ary MDS code and its dual is also MDS.
We define
(2) 
The dual of code is explicitly determined. Precisely, let
be the allone vector with appropriate length. The dual of
is , where with for .Remark 2.1.
The matrix is a generator matrix of , and is a generator matrix of . It is clear that
where is the diagonal matrix with diagonal entries .
We now introduce some basic notations and results on extended generalized ReedSolomon code. The dimensional extended code of length given by
(3) 
where is the coefficient of in .
Clearly, the code has a generator matrix
It is known that is ary MDS code and its dual is also MDS. Precisely, the dual code of is , where with for .
Remark 2.2.
Let be a generator matrix for , and let be a generator matrix for . It is easy to see that the relationship between , and is follows
where is the diagonal matrix with diagonal entries .
The following lemma gives a criterion for a code to be selfdual.
Lemma 2.1.
Let be an even integer, and . The code is a ary MDS selfdual code over if and only if for any codeword , where , we have .
Proof.
Since , for any , we have , where is a generator matrix for . By remark(2.1), we can obtain that
This implies that is MDS selfdual if and only if
In other words, . Therefore, the lemma is proved. ∎
Corollary 2.1.
([References], Corollary 2.4) For an even integer , and . If there exist such that for some for all , then the code defined in (1) is MDS selfdual, where for all .
The following lemma is important and it gives the necessary and sufficient condition to construct selfdual codes via the extended codes.
Lemma 2.2.
Let be even and . The code is a ary MDS selfdual code over if and only if for any codeword , where , we have .
Proof.
Since is a ary MDS selfdual code , we have , where is a generator matrix for , with . By remark(2.2), we get
It follows that is a ary MDS selfdual code over if and only if
That means . This completes the proof.
∎
Corollary 2.2.
([References], Lemma 2) Let be even and . If for some for all , then the code defined in (3) is MDS selfdual, where for all .
3 Main Results
The existence of MDS selfdual codes over finite fields of even characteristic has been completely addressed in [References].
In this section, we construct serval new classes of MDS selfdual codes,
by using codes and extended codes over finite fields of odd characteristic.
The following lemma can be found in [References].
Lemma 3.1.
Let be a positive integer and let be a primitive th root of unity. Then for any , we have
Let denote the set of nonzero squares of .
Theorem 1.
Let where is odd prime power, and let be a positive integer with and . Assume both and are even. Then there exists a ary MDS selfdual code over .
Proof.
Let be a primitive th root of unity. Since one has group isomorphism and embedding
and , we can choose () to be cost representatives of and
Then the entries of are distinct elements of . Note that
for any . By Lemma 3.1, for any and for any , we have
Note that since .
Since is even and is primitive th root of unity,
then . As a consequence, .
By Corollary 2.1, there exists a
ary MDS selfdual code over .
∎
Remark 3.1.
Theorem 3.4 (i) in [References] is a special case of the preceding result when .
Example 3.1.
When and , we can choose and . In this case and the preceding result shows that there exists ary MDS selfdual code of length which, as our best knowledge, has not been found in previous references.
Theorem 2.
Let where is odd prime power, and let be a positive integer with and . Assume is odd. Then there exists a ary MDS selfdual code over .
Proof.
Since is odd, then . Therefore . By Corollary 2.2, there exists a ary MDS selfdual code over .
∎
Theorem 3.
Let where is odd prime power, and let be a positive integer with and . Assume is even. Then there exists a ary MDS selfdual code over .
Proof.
Theorem 4.
Let with odd prime. For any with and , if is even, then there exists selfdual MDS code with length .
Proof.
Let be an dimensional vector subspace in satisfying .
Denote by a primitive element of order . Choose
For any ,
(4) 
where the last equality follows from that and runs through when runs through .
Denote by . Then . Since is even, we can deduce that . As a consequence, which is independent of the choice of . In this case we choose . Then . By Corollary 2.1, there exists selfdual MDS code with length . ∎
Remark 3.2.
For , the preceding result is exactly Theorem 4 (ii) in [References].
In the case is a square, usually Theorems 14 will present more classes of MDS selfdual codes than the previous results.
Example 3.2.
For , there are 243 different for which MDS selfdual code of lengths are constructed in all the previous works(in Table 1). In our constructions (Theorems 14), there are 713 different classes of MDS selfdual codes of different lengths.
Acknowledgements
This work is supported by the selfdetermined research funds of CCNU from the colleges’ basic research and operation of MOE (Grant No. CCNU18TS028). The work of J.Luo is also supported by NSFC under Grant 11471008.
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