发表机构
The Hong Kong University of Science and Technology（香港科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过引入$\mathbb{F}_q$-良好零点集并发展多种构造方法，证明了若干族原始窄感知BCH码的最小距离等于其设计距离，覆盖广泛设计距离并恢复已知结果。
AI 中文摘要
确定BCH码的精确最小距离仍然是一个未解决的问题。我们建立了若干族原始窄感知BCH码的最小距离，证明它们达到其设计距离。我们的方法集中于$\mathbb{F}_q$-良好零点集，我们通过其消逝多项式上的导数条件引入这些零点集。我们证明，长度为$q^m-1$、设计距离为$2\leq\delta\leq q^m-1$的$q$元原始窄感知BCH码具有最小距离$\delta$，当且仅当在具有$q^m$个元素的有限域$\mathbb{F}_{q^m}$中存在一个基数为$\delta+1$的$\mathbb{F}_q$-良好零点集。为了构造$\mathbb{F}_q$-良好零点集，我们开发了几种基于多项式替换、幂映射和移位逆的方法，以及使用特殊形式多项式的直接构造。结合适当的初始$\mathbb{F}_q$-良好零点集（包括那些来自已知最小距离结果的零点集），这些方法产生了各种基数的新良好零点集，从而产生了其最小距离等于其设计距离的原始窄感知BCH码族。这些族覆盖了广泛的设计距离范围，几个已知的最小距离结果作为特例被恢复。
英文摘要
Determining the exact minimum distances of BCH codes remains a open problem. We establish the minimum distances of several families of primitive narrow-sense BCH codes, showing that they attain their designed distances. Our approach centers on $\mathbb{F}_q$-good zero-sets, which we introduce through a derivative condition on their vanishing polynomials. We show that a $q$-ary primitive narrow-sense BCH code of length $q^m-1$ and designed distance $2\leqδ\leq q^m-1$ has minimum distance $δ$ if and only if there exists an $\mathbb{F}_q$-good zero-set of cardinality $δ+1$ in the finite field $\mathbb{F}_{q^m}$ with $q^m$ elements. To construct $\mathbb{F}_q$-good zero-sets, we develop several methods based on polynomial substitutions, power maps, and shifted inverses, as well as direct constructions using polynomials of special forms. Together with suitable initial $\mathbb{F}_q$-good zero-sets, including those arising from known minimum-distance results, these methods yield new good zero-sets of various cardinalities and hence families of primitive narrow-sense BCH codes whose minimum distances equal their designed distances. These families cover a broad range of designed distances, with several known minimum-distance results recovered as special cases.
