A Class of Narrow-Sense BCH Codes

TL;DR

This paper advances the understanding of narrow-sense BCH codes by determining their dimensions and weight distributions for various lengths and designed distances, including some optimal codes meeting the Griesmer bound.

Contribution

It generalizes previous results by providing explicit parameters and weight distributions for broad classes of narrow-sense BCH codes, including some optimal codes.

Findings

Dimensions of certain narrow-sense BCH codes are explicitly determined.

Weight distributions for specific classes of BCH codes are derived.

Some BCH codes are shown to be optimal with respect to the Griesmer bound.

Abstract

BCH codes are an important class of cyclic codes which have applications in satellite communications, DVDs, disk drives, and two-dimensional bar codes. Although BCH codes have been widely studied, their parameters are known for only a few special classes. Recently, Ding et al. made some new progress in BCH codes. However, we still have very limited knowledge on the dimension of BCH codes, not to mention the weight distribution of BCH codes. In this paper, we generalize the results on BCH codes from several previous papers. The dimension of narrow-sense BCH codes of length $\frac{q^m-1}{\lambda}$ with designed distance $2\leq \delta \leq \frac{q^{\lceil(m+1)/2 \rceil}-1}\lambda+1$ is settled, where $\lambda$ is any factor of $q-1$. The weight distributions of two classes of narrow-sense BCH codes of length $\frac{q^m-1}2$ with designed distance $\delta=\frac{(q-1)q^{m-1}-q^{\lfloor(m-1)/2\rfloor}-1}2$ and $\delta=\frac{(q-1)q^{m-1}-q^{\lfloor(m+1)/2\rfloor}-1}2$ are determined. The weight distribution of a class of BCH codes of length $\frac{q^m-1}{q-1}$ is determined. In particular, a subclass of this class of BCH codes is optimal with respect to the Griesmer bound. Some optimal linear codes obtained from this class of BCH codes are characterized.