New Construction of Maximum Distance Separable (MDS) Self-Dual Codes over Finite Fields.

Aixian Zhang,Zhe Ji

doi:10.3390/e21020101

Abstract

Maximum distance separable (MDS) self-dual codes have useful properties due to their optimality with respect to the Singleton bound and its self-duality. MDS self-dual codes are completely determined by the length so the problem of constructing q-ary MDS self-dual codes with various lengths is a very interesting topic. Recently X. Fang et al. using a method given in previous research, where several classes of new MDS self-dual codes were constructed through (extended) generalized Reed-Solomon codes, in this paper, based on the method given in we achieve several classes of MDS self-dual codes.

Highlights

Let Fq be the finite field with q elements
The study of Maximum distance separable (MDS) self-dual codes has attracted a great deal of attention in recent years due to its theoretical and practical importance
The center of the study of MDS codes includes the existence of MDS codes [1], classification of MDS codes [2], balanced MDS codes [3], non-Reed-Solomon MDS

Summary

Introduction

Let Fq be the finite field with q elements. A q-ary [n, k, d] linear code C is a k-dimensional subspace of with minimum (Hamming) distance d. A self-dual code is a linear code satisfying C = C ⊥. A linear complementary-dual code is a linear code satisfying C ∩ C ⊥ = {0}. The center of the study of MDS codes includes the existence of MDS codes [1], classification of MDS codes [2], balanced MDS codes [3], non-Reed-Solomon MDS codes [4], complementary-dual MDS codes [5,6], and lowest density MDS codes [7]. As the parameters of an MDS self-dual code are completely determined by the code’s length n, the main interest here is to determine the existence and give the construction of q-ary MDS self-dual codes for various lengths. Based on the method raised in [9], we present some classes of MDS self-dual codes

Preliminaries

Main Result

Conclusions