Two Classes of New MDS Self-dual Codes over Finite Fields

06/02/2019 ∙ by Xiaolei Fang, et al. ∙ Central China Normal University 0

In this paper, we produce two new classes of MDS self-dual codes via generalized Reed-Solomon (GRS) codes over finite fields. Among our constructions, for large square q, we can produce about 1/8· q new MDS self-dual codes with different lengths

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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 self-dual code.

Both MDS codes and self-dual codes have theoretical interest and practical importance. Thus the study of MDS self-dual codes have attracted a lot of interest, especially for the construction of MDS self-dual codes. MDS self-dual 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) Reed-Solomon codes, see [References, References, References, References, References, References, References].

It is noted that the parameters of MDS self-dual codes can be completely determined by the code length . For the finite field of even characteristic, Grassl and Gulliver completely determined the MDS self-dual codes of length for all in [References]. In [References, References, References], the authors obtained some new MDS self-dual codes through cyclic, constantcyclic and negacyclic codes. In [References], Jin and Xing firstly presented a general and efficient method to construct MDS self-dual codes by utilizing generalized Reed-Solomn (GRS) codes. Afterwards, many researchers start to construct MDS self-dual codes via GRS codes. In [References], Zhang and Feng gave a non-existence result, that is, when and , we can’t construct MDS self-dual code of length . We list all the known results on the systematic constructions of MDS self-dual 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
, even(except when is even, is even
and ), and
[References]
, odd , odd, and [References]
, odd
, , even, ,
even and even
[References]
, odd
, , even, ,
and
[References]
, even, odd prime with , , and [References]
, even, odd prime
with , ,
and or ,
[References]
with [References]
Table 1: Known systematic construction on MDS self-dual codes of length     ( is the quadratic character of )

In odd characteristic case, the length of MDS self-dual 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 self-dual codes with different lengths. In this paper, we produce approximately many -ary MDS self-dual codes with different lengths.

2 Preliminaries

In this section, we introduce some basic notations and useful results on self-dual codes and GRS codes.

For two vectors

and of , we can define their (Euclidean) inner product:

Let be a linear code over . The (Euclidean) dual of is defined as follows:

If , then is called a self-dual 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 well-known 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 self-dual 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 self-dual 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 .

We choose .

  • In (2), we have known that and are all nonzero square elements. From (3) and is even, we have is a nonzero square element. So .

  • In (4), from the above analysis, it is clear that all elements except are nonzero square elements. So .

In summary, by Lemma 2.1, there exists a -ary MDS self-dual code over . ∎

Remark 3.1.

In Theorem 1, the number of is and the number of is . Thus the number of MDS self-dual codes is .

Example 3.1.

For , we can construct different for which MDS self-dual code of length by utilizing all the previous results (in Table 1). But we can produce different classes of MDS self-dual codes of different lengths by Theorem 1 in this paper. So the results in this paper can construct much more MDS sel-dual 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 self-dual 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 .

We choose .

  • In (5), we have known that and are all nonzero square elements. From (6), and is odd, we obtain is a nonzero square element. So .

  • In (7), from the above analysis, it is clear that all elements except are nonzero square elements. So .

In summary, by Lemma 2.1, there exists a -ary MDS self-dual code over . ∎

Remark 3.2.

In Theorem 2, the number of is and the number of is . Thus the number of MDS self-dual codes is .

Example 3.2.

For , we can construct different for which MDS self-dual code of length by utilizing all the previous results (in Table 1). But we can produce about different classes of MDS self-dual codes of different lengths by Theorem 2 in this paper. So the results in this paper can construct much more MDS sel-dual codes than previous works for large square .

4 Conclusion

Utilizing GRS codes, we construct two new classes of MDS self-dual codes. For any large square , we can produce about new MDS self-dual 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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