Characterization and classification of binary linear codes with various hull dimensions from an improved mass formula

时间：2023年9月16日（周六）下午15:30-16:30

地点：6号楼M323

报告人简介：施敏加，博士，教授，博士生导师，数学科学学院副经理。先后入选第二届“安徽省青年数学奖”，安徽省学术与技术带头人，安徽省杰青支持计划，安徽大学英才计划、安徽大学至诚志坚拔尖人才，2020-2022年连续三年入选全球前2%顶尖科学家“年度影响力”榜单，是中国工业与应用数学学会-编码密码及相关组合理论专业委员会成员，是中科院二区 SCI 期刊 JAMC 的副主编。先后荣获安徽省自然科学一等奖和安徽省自然科学二等奖各一项、安徽省教学成果奖一等奖一项和三等奖三项、安徽省研究生导师师德标兵称号、安徽省教学名师、并获教育部宝钢优秀教师奖，主持的《近世代数》课程被认定为第二批国家一流课程。主持国家自然科学基金4项，安徽省自然科学基金杰出青年基金等省部级重点项目多项。在 Elsevier 出版社和 World Scientific 出版社出版英文学术专著 2 部，在 IEEE TIT, JCTA, DCC, FFTA 等国内外权威学术期刊上发表 SCI 期刊论文100余篇，研究成果入选《世界简明编码理论百科全书》和 ESI 高被引论文。曾应邀访问新加坡，法国，俄罗斯、韩国等国家。

报告摘要：The hull of a linear code over finite fields is the intersection of the code and its dual, which was introduced by Assmus and Key to classify finite projective planes. The main purpose of this paper is to obtain the closed mass formula for binary linear codes with various hull dimensions, which simplifies the mass formula obtained by Sendrier in (SIAM J. Discrete Math., 10(2): 282-293, 1997). We show that almost all binary linear codes with $\ell$-dimensional hull are odd-like codes with odd-like duals for fixed $\ell$. We also study the largest minimum distance of a binary linear $[n,k]$ code with $\ell$-dimensional hull. Most importantly, we give a complete classification of binary linear codes with various hull dimensions for $n\leq 12$ using a building-up construction, which is confirmed by double-checking with our mass formula. We also give the classification of optimal binary linear $[n,k]$ codes with various hull dimensions for $n\leq 13.$ Combining with known results, we obtain the classification of (optimal) binary linear codes with small parameters.