Two families of negacyclic BCH codes

TL;DR

This paper investigates negacyclic BCH codes with specific lengths over finite fields, revealing their generally good parameters and potential to outperform cyclic codes in minimum distance for certain cases.

Contribution

It provides new analyses of negacyclic BCH codes with lengths (q^m-1)/2 and (q^m+1)/2, highlighting their optimality and improved minimum distances in some scenarios.

Findings

Negacyclic BCH codes often have good parameters and include many optimal codes.

For certain q and m, these codes have larger minimum distances than cyclic codes with the same length and dimension.

The study expands understanding of negacyclic BCH codes' parameters and their advantages over cyclic codes.

Abstract

Negacyclic BCH codes are a subclass of neagcyclic codes and are the best linear codes in many cases. However, there have been very few results on negacyclic BCH codes. Let $q$ be an odd prime power and $m$ be a positive integer. The objective of this paper is to study negacyclic BCH codes with length $\frac{q^m-1}{2}$ and $\frac{q^m+1}{2}$ over the finite field $\mathbf(q)$ and analyse their parameters. The negacyclic BCH codes presented in this paper have good parameters in general, and contain many optimal linear codes. For certain $q$ and $m$, compared with cyclic codes with the same dimension and length, the negacyclic BCH codes presented in this paper have a larger minimum distance in some cases.