Problem
Research questions and friction points this paper is trying to address.
BCH codes
minimum distance
error-correcting capability
Solutions to Three Conjectures and an Open Problem on Binary BCH Codes
🤖 AI Summary
本文解决了二进制BCH码的三个猜想和一个公开问题，通过构造达到BCH界的码字确定了三类码的确切最小距离，并研究了一类更难的码的部分情况。
📝 Abstract
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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
binary BCH codes
minimum distance
BCH bound
🔎 Similar Papers
No similar papers found.
