报告题目：Edge-subset Lattice and Its Application to Linear Network Error Correction Coding
主讲人：光炫
报告时间：2024年5月19日9:00-9:30
报告地点：西安电子科技大学南校区网安大楼12楼A-1236
主讲人介绍：
光炫博士，南开大学数学科学学院教授，博士生导师，数学学科学术委员会委员；入选国家青年人才项目、香江学者计划、天津市131创新人才计划第二梯队和“三年千人”高层次人才引进计划等。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及香港中文大学从事研究工作近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，其中在信息论顶刊IEEE Trans. Inf. Theory上发表论9篇（其中8篇为双栏18页（单栏约40页）以上长文），在信息论通信理论和安全的权威期刊会议上发表论文20余篇，包括IEEE J. Sel. Areas Inf. Theory, IEEE J. Sel. Areas Commun., IEEE Trans. Inf. Foren. Sec., IEEE Trans. Commun., USENIX Security,中国科学(英文版)等。获多个会议最佳论文奖；获天津数学与统计“青年学者奖”，中国电子学会“信息论青年新星奖”等。主持重点研发计划课题和基础加强重点研究课题等基金项目8项，华为公司和中海油服物探研究院科技项目2项。
报告主要观点：
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 onlyof 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.
