A Galois Connection Approach to Wei-Type Duality Theorems

报告时间:2020年12月24日   10:00—11:00

报告地点:红瓦楼723

报告人简介:许扬 博士后

本科,2008-2012,复旦大学数学科学学院,数学与应用数学专业。

博士,2012-2017,复旦大学数学科学学院,基础数学方向。

博士后,2017-现在,复旦大学计算机科学技术学院。

博士后,2018-现在,香港大学理学院数学系。

报告摘要:In  $1991$, Wei proved a duality theorem that established an interesting  connection between the generalized Hamming weights of a linear code  and those of its dual code. Wei's duality theorem has since been  extensively studied from different perspectives and extended to other  settings. In this talk, we re-examine  Wei's duality theorem and its various extensions, henceforth referred  to as Wei-type duality theorems, from a new Galois connection  perspective. Our approach is based on the observation that the  generalized Hamming weights and the dimension/length profiles of a  linear code form a Galois connection. Our main result is a general  Wei-type duality theorem for two Galois connections between finite  subsets of $\mathbb{Z}$, from which all the known Wei-type duality  theorems can be recovered. As corollaries of the main result, we prove  new Wei-type duality theorems for some combinatorial notions, and we  further unify and generalize all the known Wei-type duality theorems  established for codes endowed with various metrics.