The dimension and Bose distance of some BCH codes of length $\frac{q^{m}-1}{\lambda}$

Abstract

BCH codes are important error correction codes, widely utilized due to their robust algebraic structure, multi-error correcting capability, and efficient decoding algorithms. Despite their practical importance and extensive study, their parameters, including dimension, minimum distance and Bose distance, remain largely unknown in general. This paper addresses this challenge by investigating the dimension and Bose distance of BCH codes of length $(q^m - 1)/\lambda$ over the finite field $\mathbb{F}_q$, where $\lambda$ is a positive divisor of $q - 1$. Specifically, for narrow-sense BCH codes of this length with $m \geq 4$, we derive explicit formulas for their dimension for designed distance $2 \leq \delta \leq (q^{\lfloor (2m - 1)/3 \rfloor + 1} - 1)/{\lambda} + 1$. We also provide explicit formulas for their Bose distance in the range $2 \leq \delta \leq (q^{\lfloor (2m - 1)/3 \rfloor + 1} - 1)/{\lambda}$. These ranges for $\delta$ are notably larger than the previously known results for this class of BCH codes. Furthermore, we extend these findings to determine the dimension and Bose distance for certain non-narrow-sense BCH codes of the same length. Several optimal linear codes can be obtained from these BCH codes.