报告名称：A tight upper bound on the number of non-zeroweights of a cyclic code

报告专家：张光辉

专家所在单位：洛阳师范学院

报告时间：2022年9月29日上午

报告地点：腾讯会议(247 361 581)

专家简介：张光辉,洛阳师范学院教授，硕士生导师，应用数学研究所所长。研究方向为代数编码理论，在《IEEE Transactions on Information Theory》、《IEEE Communications Letters》、《Designs, Codes and Cryptography》、《Finite Fields and Their Applications》、《Discrete Mathematics》、《中国科学》、《电子学报》等国内外学术期刊上发表论文40余篇，其中SCI（EI）收录30余篇。现为美国《数学评论》评论员、部分SCI期刊审稿人。

报告摘要：Let $\mathcal{C}$ be a simple-root cyclic code andlet $\mathcal{G}$ be the subgroup of the automorphism group of $\mathcal{C}$ generated by the cyclic shiftof $\mathcal{C}$ and the scalar multiplications of $\mathcal{C}$.In this paper, we find an explicit formula for the number of orbits of $\mathcal{G}$ on$\mathcal{C}\setminus\{\mathbf{0}\}$.Consequently, an explicit upper bound on the number of non-zero weights of $\mathcal{C}$ isimmediately derived and a necessary and sufficientcondition for codes meeting the bound is exhibited.Several reducibleand irreducible cyclic codesmeeting the bound are presented, revealing that our bound is tight. In particular,we find that some infinite families of irreducible cyclic codesconstructed in [Ding, IEEE-TIT, 2009] meet our bound; we then conclude that such known codes enjoy an additionalproperty that any two codewords with the same weight belong to the same $\mathcal{G}$-orbit,a fact that may not have been known before.Our main result improves and generalizes some of the results in[Shi, Li, Neri and Sol\'{e}, IEEE-TIT, 2019].