Speaker: Professor Xiang Qing
Topic: Linear Representations of Finite Geometries and Associated LDPC Codes
Date: April 17, 2023
Time: 4:00 p.m.
Venue: Academic Lecture Hall 1506, Building 9
Sponsors: School of Mathematics and Statistics, Research Institute of Mathematical Science, Institute of Science and Technology
Biography:
Xiang Qing is a Chair Professor in the Department of Mathematics at Southern University of Science and Technology. Prof. Xiang received his PhD degree from Ohio State University in 1995. His main research fields are Combinatorics, Finite Geometry, Algebraic Coding Theory and Addition Combinations. He has published 99 papers in important international journals, including the top journals in the international combinatorics world such as J. Combin. Theory Ser. A, J. Combin. Theory Ser. B and Combinatorica, as well as the top comprehensive mathematical journals Advances in Math. and Trans. Amer. Math. Soc. He has led more than 10 research projects of the National Natural Science Foundation of the United States, the National Natural Science Foundation of China, and the Joint Research Fund for Overseas Chinese, Hong Kong and Macao Young Scholars. He is leading a key project of the National Natural Science Foundation of China and a project of Overseas Senior Research Scholar Fund. He has made more than 60 reports at international academic conferences.
Abstract:
The linear representation of a subset of a finite projective space is an incidence structure of affine points and lines determined by the subset. In this talk we use character theory to show that the rank of the incidence matrix has a direct geometric interpretation in terms of certain hyperplanes. We consider the LDPC codes defined by taking the incidence matrix and its transpose as parity-check matrices, and in the former case prove a conjecture of Vandendriessche that the code is generated by words of minimum weight called plane words. In the latter case we compute the minimum weight in some cases and provide a few constructions of codewords.