AI 中文总结
本文构造无穷本原狭义BCH码反例，证明其最小距离超Bose距离，反驳Chpin猜想，该距离差随码长立方根增长且≥4，利用广义Reed-Muller码性质识别相关BCH码。
AI 中文摘要
确定BCH码的精确最小距离是一个长期存在且具有挑战性的问题。本文中，我们构造了一类无穷的本原狭义BCH码，其最小距离严格超过它们的Bose距离。设q为素数幂，m为满足m≥10且m≠12的整数，令u=⌊m/4⌋，t=⌊(m-1)/3⌋。对于每个满足u≤s<t的整数s，定义δ=qᵐ - qᵐ⁻¹ - qᵐ⁻¹⁻ᵘ - qˢ - 1。我们证明，具有设计距离δ的本原狭义BCH码的Bose距离为δ，最小距离至少为δ + qˢ，当q=2时等号成立。此外，通过令s=t-1，我们得到了一类二元BCH码子族，其中最小距离与Bose距离的差至少随码长的立方根增长，对于所有m≥13严格超过4，这反驳了Chpin的猜想。我们通过利用广义Reed-Muller码的重量可除性性质来识别这些BCH码。
英文摘要
We construct an infinite family of $q$-ary primitive narrow-sense BCH codes whose minimum distance strictly exceeds the Bose distance; in fact, the gap between the two can be arbitrarily large as the length of the code tends to infinity. The key idea is to embed these BCH codes in a suitably large punctured generalized Reed--Muller code, whose codeword weights obey divisibility conditions supplied by Ax's theorem. This divisibility forces the minimum distance of the BCH codes far above the Bose distance. In particular, our family disproves a longstanding conjecture of Charpin asserting that this difference is at most four.
