报告题目：Infinite families of 3-designs and 2-designs from almost MDS codes

主讲人：曹喜望 教授

时间：2023年6月2日（周五）16：30-17：30

地点：理学院317学习室

主办单位：理学院

主讲人简介：

曹喜望，南京航空航天大学理学院教授，博士生导师。师从樊恽教授获得硕士学位，师从北京大学丘维声教授获得博士学位。研究方向是有限域及其应用，在差集、指数和、有限域上的多项式、量子信息处理以及代数编码方面做出了出色的工作，其研究成果发表在相关领域的权威期刊上，发表学术论文150余篇。曹喜望教授先后多次访问过Sydney大学、南洋理工大学，香港科技大学、台湾中央研究院、北京国际数学中心、南开大学陈省身数学研究所等。2010年入选江苏省“青蓝工程”学术带头人。主持国家自然科学基金面上项目和省部级科研项目多项。2017年获得江苏省科学技术奖。

摘要：

Combinatorial designs are closely related to linear codes. Recently, some near MDS codes were employed to construct $t$-designs by Ding and Tang, which settles the question regarding whether there exists an infinite family of near MDS codes holding an infinite family of $t$-designs for $t \geq 2$. In this talk, I will introduce some constructions of infinite families of 3-designs and 2-designs from special equations over finite fields. First, I will present an infinite family of almost MDS codes over $ {\rm GF}(p^m)$ holding an infinite family of 3-designs. Then I will provide an infinite family of almost MDS codes over $ {\rm GF}(p^m)$ holding an infinite family of 2-designs  for any field ${\rm GF}(q)$. In particular, some of these almost MDS codes are near MDS. Second, I will present an infinite family of near MDS codes over ${\rm GF}(2^m)$ holding an infinite family of 3-designs by considering the number of roots of a special linearized polynomial. Compared to previous constructions of 3-designs or 2-designs from linear codes, the parameters of some of our designs are new and flexible. This is a joint work with Guangkui Xu and Longjiang Qu.