The Exact Parameters of A Family of BCH Codes

TL;DR

This paper precisely determines the minimum distance and dimension of a specific family of BCH codes, resolving three open problems and advancing the theoretical understanding of these codes.

Contribution

It establishes exact parameters for narrow-sense BCH codes, solving longstanding open problems and building on previous theoretical frameworks.

Findings

Exact minimum distance and dimension formulas derived

Resolves three open problems in BCH code theory

Advances understanding of BCH code parameters

Abstract

Despite the theoretical and practical significance of BCH codes, the exact minimum distance and dimension remain unknown for many families. This paper establishes the precise minimum distance and dimension of narrow-sense BCH codes $\C_{(q, m, \lambda, \ell_0, \ell_1)}$ over $\gf(q)$ of length $\frac{q^m-1}{\lambda}$ and designed distance $\frac{(q-\lambda \ell_0)q^{m-1-\ell_1}-1}{\lambda}$, where $\lambda\mid (q-1)$, $0\leq \ell_0< \frac{q-1}{\lambda}$, and $0\leq \ell_1\leq m-1$. These results conclusively resolve the three open problems posed by Li et al. (IEEE Trans. Inf. Theory, vol. 63, no. 11, pp. 7219-7236, Nov. 2017) while establishing complementary advances to Ding's seminal framework (IEEE Trans. Inf. Theory, vol. 61, no. 10, pp. 5322-5330, Oct. 2015).