Explicit MDS Codes With Complementary Duals

Abstract

In 1964, Massey introduced a class of codes with complementary duals which are called Linear Complimentary Dual (LCD for short) codes. He showed that LCD codes have applications in communication system, side-channel attack (SCA) and so on. LCD codes have been extensively studied in literature. On the other hand, MDS codes form an optimal family of classical codes which have wide applications in both theory and practice. The main purpose of this paper is to give an explicit construction of several classes of LCD MDS codes, using tools from algebraic function fields. We exemplify this construction and obtain several classes of explicit LCD MDS codes for the odd characteristic case.

Several constructions of LCD MDS codes

“…Beelen 和 Jin [40] 利用函数域和代数几何码的框架, 重新描述和构造了 LCD MDS 码. 研究结果表 明, 函数域的全部机制都可以用于研究 LCD MDS 码, 代数几何框架也有助于对 GRS 码进行更精细 的分析.…”

“…研究结果表 明, 函数域的全部机制都可以用于研究 LCD MDS 码, 代数几何框架也有助于对 GRS 码进行更精细 的分析. 限于篇幅, 本小节只介绍其主要结果, 更详细的内容, 可参见文献 [40].…”

On MDS negacyclic LCD codes

“…So, the following is immediate. 22,7] , [28, 20,9] , [28, 18,11] , [28, 16,13] , [28, 14,15] , [14,12,3] , [14,10,5] , [14,8,7] , [14,6,9] , [14,4,11] , [14,2,13] , [7,6,2] , [7,4,4] , [7,2,6] , [4,3,2] .…”

“…In [6], Chen and Liu have proposed a different approach to obtain new LCD MDS codes from generalized Reed-Solomon codes. In [2], Beelen and Jin gave an explicit construction of several classes of LCD MDS codes, using tools from algebraic function fields. In [4], Carlet et al have studied several constructions of new Euclidean and Hermitian LCD MDS codes.…”

MDS Multi-twisted Reed-Solomon codes with small dimensional hull

“…In [22], X. Yang and J. L. Massey obtained a necessary and sufficient condition for a cyclic code C of length n to be an LCD code which is that the generator polynomial g(x) of C be self-reciprocal and all the monic irreducible factors of g(x) have the same multiplicity in g(x) as in x n − 1. For other constructions of LCD codes we refer to [2] and [6]. Moreover, H. Liu and S. Liu studied MDS LCD multi-twisted RS codes in [11].…”