达到设计距离的广义BCH码和扭曲戈帕码
Generalized BCH Codes and Twisted Goppa Codes Attaining Their Designed Distances
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中文总结 AI 辅助
研究广义BCH码和扭曲戈帕码最小距离，通过校验矩阵给出替代码达设计距离的充要条件并用于广义BCH码，证明多类广义BCH码最小距离等于设计距离，刻画特定扭曲戈帕码满足某距离的情况并推导相关族。
中文摘要 AI 辅助
在编码理论中，确定替代码的真正最小距离一直是个极其困难的问题。本文通过校验矩阵研究广义BCH码和扭曲戈帕码的最小距离。首先给出替代码达到其设计距离的充要条件并应用于广义BCH码，证明了广义BCH码的广泛类别具有等于其设计距离的最小距离，给出了明确的无限族。还刻画了\(\operatorname{deg} g=t\)的扭曲戈帕码\(\Gamma(L,g,\eta)\)满足\(d(\Gamma(L,g,\eta))=t + 1\)的情况，并推导了达到此距离的结构化类和无限族。
英文摘要
Determining the true minimum distance of an alternant code remains a notoriously difficult problem in coding theory. In this paper, we study the minimum distances of generalized BCH codes and twisted Goppa codes through their parity-check matrices. We first give a necessary and sufficient condition for an alternant code to attain its designed distance and apply it to generalized BCH codes. As applications, we prove that broad classes of generalized BCH codes have minimum distances equal to their designed distances. These classes provide explicit infinite families rather than isolated examples. We characterize when a twisted Goppa code $Γ(L,g,η)$ with $\operatorname{deg} g=t$ satisfies $d(Γ(L,g,η))=t+1$, and derive structured classes and infinite families attaining this distance.
