In spite of the intensive study of cyclic codes and the recent construction of an infinite family of self-dual binary cyclic codes whose minimum distances have the square-root bound in IEEE Trans. IT, vol. 71, no. 4, 2025, it is still a 70-year-old open problem whether there is an infinite family of self-dual binary cyclic codes whose minimum distances have a lower bound better than the square-root bound. This paper settles this long-standing open problem in coding theory by presenting infinite families of such self-dual binary cyclic codes. As by-products, several families of cyclic codes with better parameters than those in some references are also constructed in this paper.

翻译：尽管循环码研究已非常深入，且近期在《IEEE信息论汇刊》第71卷第4期（2025年）中构造了极小距离具有平方根界的无限族自对偶二元循环码，但关于是否存在极小距离下界优于平方根界的无限族自对偶二元循环码，仍是困扰学界长达70年的公开问题。本文通过构造此类自对偶二元循环码的无限族，解决了这一编码理论中的长期公开难题。作为副产品，本文还构造了若干参数优于部分参考文献的循环码族。