Euclidean and Hermitian LCD MDS codes

TL;DR

This paper investigates the existence and construction of Euclidean and Hermitian LCD MDS codes, providing complete solutions for their existence over various finite fields and introducing new construction methods.

Contribution

It completely solves the existence problem for Euclidean LCD MDS codes over finite fields and proposes new construction techniques for both Euclidean and Hermitian LCD MDS codes.

Findings

Existence of q-ary [n,k] Euclidean LCD MDS codes for 0 ≤ k ≤ n ≤ q+1 when q > 3.

Existence of q-ary [n,k] Euclidean LCD MDS codes with n=q+2 and k=3 or q-1 for q=2^m.

New constructions of Euclidean and Hermitian LCD MDS codes using self-orthogonal and Reed-Solomon codes.

Abstract

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.