Counterexamples to Charpin's Conjecture on BCH codes

Abstract

Determining the exact minimum distance of BCH codes is a longstanding and challenging problem. In this paper, we construct an infinite family of primitive narrow-sense BCH codes whose minimum distance strictly exceeds their Bose distance. Let $q$ be a prime power, let $m$ be an integer with $m \geq 10$ and $m \neq 12$, and set $u = \lfloor m/4 \rfloor$ and $t = \lfloor (m-1)/3 \rfloor$. For each integer $s$ with $u \leq s < t$, we define$$\delta = q^m - q^{m-1} - q^{m-1-u} - q^s - 1.$$We prove that the primitive narrow-sense BCH code with designed distance $\delta$ has Bose distance $\delta$ and a minimum distance of at least $\delta + q^s$, with equality holding for $q = 2$. Furthermore, by setting $s = t - 1$, we derive a subfamily of binary BCH codes in which the gap between the minimum distance and the Bose distance grows at least as the cube root of the code length, strictly exceeding $4$ for all $m \geq 13$. This disproves Charpin's conjecture. We identify these BCH codes by exploiting the weight divisibility properties of generalized Reed--Muller codes.