On MDS Negacyclic LCD Codes

TL;DR

This paper presents new constructions of maximum distance separable (MDS) negacyclic LCD codes over finite fields, expanding the known families and providing parameters including dimensions and bounds on minimum distances.

Contribution

The paper introduces four new families of MDS negacyclic LCD codes and several Hermitian MDS negacyclic LCD codes, with explicit parameters and bounds, advancing the construction methods in coding theory.

Findings

Constructed four families of MDS negacyclic LCD codes with specific lengths.

Determined dimensions and minimum distance bounds for various code families.

Provided new lower bounds on minimum distances using cyclotomic class analysis.

Abstract

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.