Counterexamples to Charpin's Conjecture on BCH codes

2026-07-30

AI Summary

This work addresses the long-standing problem of determining the exact minimum distance of BCH codes, a central question in coding theory closely tied to the validity of Charpin’s conjecture. By constructing an infinite family of primitive narrow-sense BCH codes with carefully chosen designed distances and leveraging weight divisibility properties of generalized Reed–Muller codes together with algebraic combinatorial techniques, the authors establish the first infinite family of counterexamples wherein the true minimum distance strictly exceeds the Bose distance. Notably, for the binary case (q = 2), the minimum distance is precisely δ + 2^s, and when the code length parameter m ≥ 13, this gap surpasses 4. These results definitively disprove Charpin’s conjecture and substantially advance the understanding of minimum distance behavior in 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$$δ= q^m - q^{m-1} - q^{m-1-u} - q^s - 1.$$We prove that the primitive narrow-sense BCH code with designed distance $δ$ has Bose distance $δ$ and a minimum distance of at least $δ+ q^s$, 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.

Problem

• BCH codes
• minimum distance
• Bose distance
• Charpin's conjecture
• primitive narrow-sense

Innovation

• BCH codes
• minimum distance
• Bose distance
• Charpin's conjecture
• generalized Reed–Muller codes