发表机构
Hubei University; Nanchang Hangkong University（湖北大学; 南昌航空大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对Chen等人提出的二元BCH码的三个猜想和一个公开问题，通过构造码字达到BCH界解决了前三个猜想，还确定了部分码族的最小距离，对公开问题8.4给出了部分肯定结果及反例。
AI 中文摘要
BCH码是最重要的循环码类别之一，在编码理论及其应用中发挥了核心作用。研究BCH码的基本问题之一是确定它们的精确最小距离，这直接决定了它们的纠错能力。尽管BCH界提供了一个通用下界，但确定精确最小距离通常很困难，许多参数族仍未解决。本文研究了Chen、Xie和Ding在文献\uc17b{Chen59}中提出的关于二元BCH码的三个猜想和一个公开问题。我们通过构造达到BCH界的码字，解决了这三个二元BCH码族的精确最小距离猜想。除了这些猜想，我们还进一步研究了更困难的码族，并确定了其在某些情况下的最小距离。我们还研究了公开问题8.4：前两个长度族得到了肯定回答，而对于第三个族，建立了一个充分条件，且存在一个反例表明无限制的断言通常不成立。
英文摘要
BCH codes are among the most important classes of cyclic codes and have played a central role in coding theory and its applications. One of the fundamental problems in the study of BCH codes is to determine their exact minimum distances, which directly govern their error-correcting capability. Although the BCH bound provides a general lower bound, determining the exact minimum distance is often difficult, and many parameter families remain unresolved. In this paper, we investigate three conjectures and an open problem on binary BCH codes proposed by Chen, Xie, and Ding in \cite{Chen59}. We settle these conjectures on the exact minimum distances of three families of binary BCH codes by constructing codewords attaining the BCH bound. Beyond these conjectures, we further study the more difficult family codes and determine its minimum distance for some cases. We further study Open Problem 8.4: affirmative answers are obtained for the first two length families, while for the third family a sufficient condition is established and a counterexample shows that the unrestricted assertion does not hold in general.
