The reversible negacyclic codes over finite fields

TL;DR

This paper investigates the structure and parameters of reversible negacyclic codes over finite fields, deriving new code constructions, including optimal and MDS LCD negacyclic codes, based on cyclotomic cosets and polynomial factorizations.

Contribution

It provides a detailed analysis of the structure of reversible negacyclic codes and introduces new classes of optimal and MDS LCD negacyclic codes with explicit parameters.

Findings

Parameters of negacyclic BCH codes for specific lengths

Construction of some optimal linear negacyclic codes

Discussion of a class of MDS LCD negacyclic codes

Abstract

In this paper, by investigating the factor of the $x^n+1$, we deduce that the structure of the reversible negacyclic code over the finite field $\mathbb{F}_{q}$, where $q$ is an odd prime power. Though studying $q-$cyclotomic cosets modulo $2n$, we obtain the parameters of negacyclic BCH code of length $n=\frac{q^\ell+1}{2}$ , $n=\frac{q^m-1}{2(q-1)}$ and $n=\frac{q^{t\cdot2^\tau}-1}{2(q^t+1)}$. Some optimal linear codes from negacyclic codes are given. Finally, we discuss a class of MDS LCD negacyclic codes.