On MDS Negacyclic LCD Codes

Linear codes with complementary duals (LCD) have a great deal of significance amongst linear codes. Maximum distance separable (MDS) codes are also an important class of linear codes since they achieve the greatest error correcting and detecting capabilities for fixed length and dimension. The construction of linear codes that are both LCD and MDS is a hard task in coding theory. In this paper, we study the constructions of LCD codes that are MDS from negacyclic codes over finite fields of odd prime power $q$ elements. We construct four families of MDS negacyclic LCD codes of length $n|\frac{{q-1}}{2}$, $n|\frac{{q+1}}{2}$ and a family of negacyclic LCD codes of length $n=q-1$. Furthermore, we obtain five families of $q^{2}$-ary Hermitian MDS negacyclic LCD codes of length $n|\left( q-1\right)$ and four families of Hermitian negacyclic LCD codes of length $n=q^{2}+1.$ For both Euclidean and Hermitian cases the dimensions of these codes are determined and for some classes the minimum distances are settled. For the other cases, by studying $q$ and $q^{2}$-cyclotomic classes we give lower bounds on the minimum distance

Explicit MDS Codes with Complementary Duals

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

Constructions of optimal LCD codes over large finite fields

In this paper, we prove existence of optimal complementary dual codes (LCD codes) over large finite fields. We also give methods to generate orthogonal matrices over finite fields and then apply them to construct LCD codes. Construction methods include random sampling in the orthogonal group, code extension, matrix product codes and projection over a self-dual basis.Comment: This paper was presented in part at the International Conference on Coding, Cryptography and Related Topics April 7-10, 2017, Shandong, Chin

Systems of MDS codes from units and idempotents

Algebraic systems are constructed from which series of maximum distance separable (mds) codes are derived. The methods use unit and idempotent schemes

Euclidean and Hermitian LCD MDS codes

Linear codes with complementary duals (abbreviated LCD) are linear codes whose intersection with their dual is trivial. When they are binary, they play an important role in armoring implementations against side-channel attacks and fault injection attacks. Non-binary LCD codes in characteristic 2 can be transformed into binary LCD codes by expansion. On the other hand, being optimal codes, maximum distance separable codes (abbreviated MDS) have been of much interest from many researchers due to their theoretical significant and practical implications. However, little work has been done on LCD MDS codes. In particular, determining the existence of $q$-ary $[n,k]$ LCD MDS codes for various lengths $n$ and dimensions $k$ is a basic and interesting problem. In this paper, we firstly study the problem of the existence of $q$-ary $[n,k]$ LCD MDS codes and completely solve it for the Euclidean case. More specifically, we show that for $q>3$ there exists a $q$-ary $[n,k]$ Euclidean LCD MDS code, where $0\le k \le n\le q+1$, or, $q=2^{m}$, $n=q+2$ and $k= 3 \text{or} q-1$. Secondly, we investigate several constructions of new Euclidean and Hermitian LCD MDS codes. Our main techniques in constructing Euclidean and Hermitian LCD MDS codes use some linear codes with small dimension or codimension, self-orthogonal codes and generalized Reed-Solomon codes