Two types of narrow-sense negacyclic BCH codes

TL;DR

This paper investigates two specific types of negacyclic BCH codes, providing their dimensions, minimum distance bounds, and weight distributions for certain parameters, advancing understanding of their properties.

Contribution

It introduces new results on the dimensions, minimum distances, and weight distributions of two classes of negacyclic BCH codes with specific lengths.

Findings

Dimensions of the codes are determined.

Lower bounds on minimum distances are established.

Weight distributions are derived for certain parameters.

Abstract

Negacyclic BCH codes are an important subclass of negacyclic codes and are the best linear codes in most cases, but their parameters are hard to determine. In this paper, we mainly study two types of negacyclic BCH codes of length $n=\frac{q^{m}-1}{4},\frac{q^{m}+1}{4}$, and give their dimensions and the lower bound on their minimum distance. Furthermore, we provide the weight distribution of narrow-sense neagcyclic BCH codes of length $n=\frac{q^m-1}{4}$ for some special designed distances.