An Unified Approach on Constructing of MDS Self-dual Codes via Reed-Solomon Codes

05/16/2019 ∙ by Aixian Zhang, et al. ∙ Tsinghua University 0

Based on the fundamental results on MDS self-dual codes over finite fields constructed via generalized Reed-Solomon codes JX and extended generalized Reed-Solomon codes Yan, many series of MDS self-dual codes with different length have been obtained recently by a variety of constructions and individual computations. In this paper, we present an unified approach to get several previous results with concise statements and simplified proofs, and some new constructions on MDS self-dual codes. In the conclusion section we raise two open problems.

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1. Introduction

Let be a linear code with parameters Namely, is a -subspace of with dimension and the minimum (Hamming) distance We have the Singleton bound is called MDS code if The dual code of is defined by

where for and is the usual inner product in . is a linear code with parameters where If is called self-dual. For self-dual code , the length of the code should be even by

MDS self-dual codes have attracted a lot of attention in recent years by their theoretical interests in coding theory, many applications in cryptography and combinatorics. For such codes, and are determined by the length . One of the basic problems is that for a fixed prime power which even number can be the length of an MDS self-dual code over finite field ?

This problem has been solved in the case by Grassl and Gulliver [5]. From now on, we consider where

is an odd prime number and

. Several families of MDS self-dual codes over have been constructed with length satisfying certain conditions by using generalized Reed-Solomon (GRS for short) codes and extended generalized Reed-Solomon (EGRS for short) codes [2]-[4],[7],[9]-[12],[15], orthogonal designs [8, 14], extended cyclic duadic codes and negacyclic codes [6]. Roughly speaking, the first approach is to look for the GRS codes and EGRS codes as candidates of MDS codes, then to find sufficient conditions satisfied by length such that the codes are self-dual. The last two approaches are to look for the self-dual codes given by orthogonal designs and (nega-)cyclic codes and select ones being MDS codes. A table of MDS self-dual codes over is provided in [14] for length and odd prime number

In this paper, we focus on the first approach to consider the MDS codes given by GRS and EGRS codes. L.Jin, C.Xing [9] and H.Yan [15] present basic results on necessary and sufficient conditions for the GRS codes and EGRS codes being self-dual respectively. By using these conditions, several families of MDS self-dual codes with various length have been constructed [2, 3, 9, 12, 15] with careful computations.

In Section 2, we introduce the basic results given in [9] and [15], but we provide an unified point of view on the constructions given there. In Section 4, we select some known results stated and proved with our approach. In Section 5, we construct new MDS self-dual codes with our approach. In Section 6, we make conclusion and raise two open problems.

2. Constructions of MDS Self-dual codes via GRS Codes

2.1. (Extended) Generalized RS Codes

Definition 2.1.

Let be a subset of with distinct elements (so that ), be nonzero elements in (not necessarily distinct), For the GRS code is defined by

This is an MDS (linear) code over with parameters The extended GRS code is defined by

where is the coefficient of in This is also an MDS code over with parameters

A sufficient condition on set has been given in [9] and [15] for and being self-dual. From the proofs we can see that the sufficient condition is also necessary.

For we denote

Let be the quadratic (multiplicative) character of Namely, for ,

Now we introduce the basic results given in [9] and [15].

Theorem 2.2.

Let be distinct elements in

(1)  ([9]) Suppose that is even. There exists such that the (MDS) code is self-dual if and only if all are the same.

(2) ([15]) Suppose that is odd. There exists such that the (MDS) code is self-dual code with length if and only if for all

Definition 2.3.

Let be the set of all even number such that there exists MDS self-dual code over with length . Let and be the set of all even number such that there exists MDS self-dual code over with length constructed by generalized RS code (Theorem 2.2 (1)) and extended generalized RS code (Theorem 2.2 (2)) respectively. Namely,

We have As a direct consequence of Theorem 2.2, we have

Corollary 2.4.

([9]) Suppose that and where is an odd prime number and Then for any even number if and if Therefore for all even number

Proof.

Suppose that Let be any subset of with size Then for each Since is the quadratic extension of each non-zero element of is a square in Therefore for all By Theorem 2.2 (1), we get Similarly we get that for even number by Theorem 2.2 (2) and taking a subset of with size

3. Binomials and Linearized Polynomials

In general case, we should choose a subset of and compute carefully to verify the conditions stated in Theorem 2.2. The following simple fact can be used to simplify the computation of in many cases. For a polynomial we denote the derivative of by

Lemma 3.1.

(1) Let be a subset of Then for any

(2) Let and be disjoint subsets of Then for

(3) Let be a monic polynomial of degree in Assume that for each has exactly distinct solutions in Namely, the size of is for all Let Then for we have where

Proof.

(1) From we get, for

(2) From we get

For and then

Similarly, for

(3) For is the set of zeros of Namely, We get

where Then from we get, for

Let . In many known results the set is chosen as a subgroup of where or a coset of in or an union of coset For is a binomial and Another candidate of is -subspace of where is a subfield of a coset or an union of cosets. It is well-known that

is a -linearized polynomial in and where

Definition 3.2.

Let be a subfield of A -linearized polynomial in has the following form

(3.1)

Let be a -linearized polynomial in . The mapping

is -linear. Namely, for

Thus the kernel and image of

are -subspaces of

For more facts on linearized polynomials we refer to the book [13], Section 3.4.

4. Review on Some Known Results

In this section, we select few known results on MDS self-dual codes constructed via GRS codes and EGRS codes. We present simple proofs with the approach illustrated in Section II. For more complete list of known MDS self-dual codes we refer to the table in [3] and [15].

Theorem 4.1.

([3]) Let and Then for each if is even and if is odd.

Proof.

Let for Then is a subfield of and From we know that We take a -subspace of with Then and

is a -linearized polynomial in Take then For any Therefore are distinct cosets of in We take a subset of with size Let and Then and

where Thus for each and we have and by Lemma 3.1 (3),

where and then

where . Since we have and If is even, is the same for all From Theorem 2.2 (1) we get If is odd, then is even and then for all From Theorem 2.2 (2), we get

Corollary 4.2.

([15]) Let Then for all even and we have

Proof.

Take in Theorem 4.1. ∎

In Theorem 4.1, is chosen as an union of cosets of a subspace of Now we consider being an union of cosets of a (cyclic) subgroup of Let then is a subgroup of with All cosets of in are the -th cyclotomic classes

Let be distinct cosets of in Then

(4.1)

where

In [3], is taken as a subset of for Firstly we should determine that how many elements in can be taken such that the cosets are distinct.

Lemma 4.3.

Let where and Let be a subset of where are distinct module Then are distinct cosets of in if and only if are distinct module Particularly, if there exist elements in such that are distinct.

Proof.

For

But and

Therefore if and only if

Now suppose that be a subset of such that are distinct module Let Then

(4.2)

is an union of cosets of in and For each by (4.1), we get

(4.3)

From we get

where Therefore where

and then

(4.4)

where and

(4.5)

Since and from (4.3),  (4.4) and (4.5), we get

(4.6)

Moreover, let Then and for By (4.3),  (4.6) and we get

(4.7)

For

and

(4.8)

With above preparation we introduce the following result which appear in [3] but with slight different statement.

Theorem 4.4.

Let be a power of a prime number and

(I) Assume that is even and