New Parameters on MDS Self-dual Codes over Finite Fields

11/07/2018 ∙ by Xiaolei Fang, et al. ∙ Central China Normal University 0

In this paper, we produce new classes of MDS self-dual codes via (extended) generalized Reed-Solomon codes over finite fields of odd characteristic. Among our constructions, there are some new parameters on MDS self-dual code which have never been reported.

READ FULL TEXT VIEW PDF
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

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 self-dual code. Self-dual codes have important applications in coding theory [References], cryptograph [References, References], combinatorics [References, References] and other related areas.

MDS self-dual codes have good properties due to its optimality with respect to the Singleton bound and its self-duality, which have attracted a lot of attentions in recent years. Some people constructed MDS self-dual codes through the way of orthogonal designs, see [References, References, References, References]. Some people utilized constacyclic codes to construct MDS self-dual codes, see [References, References, References, References]. Some people make the construction by Reed-Solomon codes, generalized Reed-Solomon codes or extended generalized Reed-Solomon codes, see [References, References, References, References, References].

Parameters of MDS self-dual codes are completely determined by their lengths , that is, . Therefore, the problem for constructing different MDS self-dual codes can be transformed to find MDS self-dual 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 self-dual codes with different even lengths possibly exist assuming MDS conjecture is valid. But up to now, only 702 MDS self-dual codes of different even lengths are able to be constructed. In [References], Jin and Xing constructed some classes of new MDS self-dual codes through generalized Reed-Solomon codes. In [References], Yan generalized the technique in [References] and constructed several classes of MDS self-dual codes via generalized Reed-Solomon codes and extended generalized Reed-Solomon codes. In [References], Labad, Liu and Luo produced more classes of MDS self-dual codes based on [References] and [References]. All the known results on the construction of MDS self-dual 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 self-dual code. Among our constructions, there are several MDS self-dual codes with new parameters (see Table 2). In particular, for square , we can produce much more MDS self-dual 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 Reed-Solomon codes. In Section 3, we will present our main results on the constructions of MDS self-dual 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]
Table 1: Known results on MDS self-dual codes of length     ( is the quadratic character of )
even Reference
, odd , , even Theorem 1 (i)

, odd
, even(expect is even, is even
and ), and
Theorem 1 (i)
, odd , odd, and Theorem 2
, odd
, , even, ,
even and even
Theorem 3 (i)
, odd
, , even, ,
and
Theorem 3 (ii)
, odd prime , Theorem 4
, odd prime , and , even Theorem 5
Table 2: Our results

2 Preliminaries

In this section, we introduce some basic notations and useful results on (extended) generalized Reed-Solomon 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 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 with such that the code defined in (1) is an MDS self-dual code of length .

Moreover, extended code can also be applied into the construction of MDS self-dual 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 well-known 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 self-dual 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 self-dual 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 self-dual code.

(ii). Assume . There exists a -ary MDS self-dual 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 self-dual 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 self-dual code.

Therefore, we know that there exists a -ary MDS self-dual code with the length of .

(ii). Similarly as (i), we let

We will find such that is an MDS self-dual 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 self-dual 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 self-dual code of length . This is a new parameter of MDS self-dual 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 self-dual 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 self-dual 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 self-dual code of length . This is a new parameter of MDS self-dual 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 self-dual code.

(ii). Assume . There exists a -ary MDS self-dual 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 self-dual 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 .

Choose with . Define

Then by Lemma 2.1, is an MDS self-dual code.

Therefore, we know that there exists a -ary MDS self-dual code with the length of .

(ii). Similarly as (i), we let

We will find such that is an MDS self-dual 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 self-dual 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 self-dual code of length . This MDS self-dual 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 self-dual code of length of , where .

Proof.

Denote by . Let be an -dimensional

-vector subspace of

, with . Choose , such that . Let , and .