报告名称：Several families of binary minimal linear codes with few weights from two-to-one functions

报告专家：曹喜望

专家单位：南京航空航天大学

报告时间：9月26日14：30

报告地点：双创大楼10楼A1009会议室

专家简介：曹喜望，南京航空航天大学理学院教授，博士生导师。北京大学获得博士学位。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期刊IEEE Transaction on Infor-mation Theory、Finite Fields and their Applications、Design Codes and Cryptography、Science China（Mathematics）等，发表学术论文近200篇，其中SCI检索论文170余篇。曹喜望教授先后多次访问过悉尼大学、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。主持完成国家自然科学基金项目5项和省部级科研项目多项。2017年获得江苏省科学技术奖。

报告摘要：Minimal linear codes have important applications in secure communications, including in the framework of secret sharing schemes and secure multi-party computation. A lot of research have been carried out to derive codes with few weights (but more importantly, being minimal) using algebraic or geometric approaches. One of the main power and fructify algebraic methods is based on the design of those codes by employing functions over finite fields. Li et al. (2021) have recently identified some binary linear codes with few weights from two classes of two-to-one functions. In this paper, our ultimate objective is to expand the class of codes derived from the paper of Li et al. by proposing larger classes of binary linear codes with few weights via ge-neric constructions involving other known families of two-to-one functions over the finite field F_{2^n} of order 2^n. We succeed in constructing such codes, and we also completely determine their weight distributions. The linear codes presented in this paper differ in parameters from those known in the literature. Besides, some of them are optimal concerning the well-known Griesmer bound. Notably, we prove that our codes are either optimal or almost optimal with re-spect to the online Database of Grassl. We next observe that the derived binary linear codes also have the minimality property for most cases. We then describe the access structures of the secret-sharing schemes based on their dual codes. Finally, we solve two problems left open in the paper by Li et al. (more specifically, a complete solution to Problem 2 and a partial solution to Problem 1).