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 $m\ge 10$ and $m\ne 12$, and set $u=\lfloor m/4\rfloor$ and $t=\lfloor (m-1)/3\rfloor$. For each integer $s$ with $u\le s<t$, 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 $m\ge 13$. This disproves Charpin's conjecture. We identify these BCH codes by exploiting the weight divisibility properties of generalized Reed--Muller codes.