报告题目：Edge-subset Lattice and Its Application to Linear Network Error Correction Coding

报 告 人：光炫教授（南开大学）

报告时间：2025年1月9日上午10:00-11:30

报告地点：广东工业大学龙洞校区行政楼610

主 持 人：陈智奇

摘要：In this talk, we first explore the underlying mathematical structure of edge subsets on a finite directed acyclic graph in using a lattice-theoretic approach. We prove that a collection of edge subsets with certain conditions, associated with the corresponding “cut-separating” partial order, forms a (semi-)lattice. The bottom and top thus derived generalize the concept of the primary minimum cut introduced by Guang and Yeung (2018) and hence we provide a new way from a lattice-theoretic point of view to understand the primary minimum cut and to justify its existence and uniqueness. We further develop efficient algorithms to find the top and the bottom, whose computational complexity is in a linear time of the number of edges in the graph. The introduced concepts and obtained results regarded as a bridge connect graph theory and lattice theory, which appear to be of fundamental interest in graph theory, lattice theory, and even beyond. In addition, by applying the approach of the edge-subset (semi-)lattice, we obtain an improved upper bound on the minimum required field size for the existence of linear network error correction (LNEC) codes. In LNEC coding, the minimum required field size for the existence of LNEC (MDS) codes is an open problem not only of theoretical interest but also of practical importance. We quantify the improvement over the existing results by both theoretical analysis and numerical simulations and thus show that the improvement is in general significant.

光炫博士，南开大学数学科学学院教授，博士生导师，副院长，南开大学数学学科学术委员会委员、教育部“核心数学与组合数学”重点实验室固定研究人员；入选国家青年人才项目、香江学者计划和南开大学百名青年学科带头人培养计划（A类）。2012年毕业于南开大学陈省身数学研究所，获博士学位，曾在美国南加州大学信息科学研究所及香港中文大学网络编码研究所从事研究工作近5年。光炫博士的研究兴趣为信息论、编码理论与密码学；目前的研究方向为面向函数计算的信息论和编码。近年来出版一部学术专著（一作），由德国Springer出版；发表学术论文60余篇，其中在信息论、安全和通信理论的权威期刊和会议上发表论文30余篇，包括 IEEE Transactions on Information Theory, IEEE Journal on Selected Areas in Information Theory, IEEE Journal on Selected Areas in Communications, IEEE Transactions on Information Forensics and Security, IEEE Transactions on Communications, USENIX Security,《中国科学》等。其研究成果获多个国内外会议的最佳论文奖。2021获天津数学与统计联合学术年会“青年学者奖”；2018 年获得中国电子学会“信息论青年新星奖”；2018年入选天津市“131创新人才计划”第二梯队、2016 年入选“香江学者计划”，2014年入选天津市“三年千人”高层次人才引进计划等。主持重点研发计划课题和基础加强重点研究课题等省部级基金项目8项，企业科技项目2项，获田家炳教育基金资助。