2026/07/30 by Run Zheng, Yaoran Yang, Yutong Zhang +1
Computer Science · Mathematics · #cs.IT #math.IT #msc:94B15
arxiv created 2026/07/30 · arxiv updated 2026/08/03
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 ≥ 10 and m ≠ 12, and set u = \lfloor m/4 \rfloor and t = \lfloor (m-1)/3 \rfloor. For each integer s with u ≤ s < t, we defineδ= qm - qm-1 - qm-1-u - qs - 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+ qs, 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 ≥ 13. This disproves Charpin's conjecture. We identify these BCH codes by exploiting the weight divisibility properties of generalized Reed--Muller codes.