报 告 人：唐春明 研究员
The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-studied subclass of cyclic codes that have found numerous applications in error correction and notably in quantum information processing. They are widely used in data storage and communication systems. A subclass of attractive BCH codes is the narrow-sense BCH codes over the Galois field GF(q) with length q+1, which are closely related to the action of the projective general linear group of degree two on the projective line. Despite its interest, not much is known about this class of BCH codes. In this talk we aim to study some of the codes with in this class and specifically narrow-sense antiprimitive BCH codes (these codes are also linear complementary duals (LCD) codes that have interesting practical recent applications in cryptography, among other benefits).We shall use tools and combine arguments from algebraic coding theory, combinatorial designs, and group theory (group actions, representation theory of finite groups, etc.) to investigate narrow-sense antiprimitive BCH Codes and extend results from the recent literature. Notably, the dimension, the minimum distance of some q-ary BCH codes with length q+1, and their duals are determined. The dual codes of the narrow-sense antiprimitive BCH codes derived include almost MDS codes. Furthermore, the classification of PGL(2,pm)-invariant codes over GF(ph) is completed. As an application of this result, the p-ranks of all in cadence structures invariant under the projective general linear group PGL(2,pm) are determined. Furthermore, infinite families of narrow-sense BCH codes admitting a 3- transitive automorphism group are obtained. Via these BCH codes, a coding theory approach to constructing the Witt spherical geometry designs is presented. The BCH codes proposed in this talk are good candidates for permutation decoding, as they have a relatively large group of automorphisms.
唐春明，博士，西华师范大学数学与信息学院研究员, 2021年度布尔奖（George Boole Prize）的杰出青年学者奖获得者。博士毕业于北京大学，先后在巴黎第八大学和香港科技大学从事研究工作。主要研究包括密码、编码及其相关的数学理论。主持国家级和省部级项目多项，在国内外重要学术期刊如《IEEE Transactions on Information Theory》、《Finite Fields and Their Applications 》、《Designs, Codes and Cryptography 》与《Science China》等发表六十多篇论文。目前担任编码与通信领域国际学术期刊《Cryptography and Communications》、《Advances in Mathematics of Communications》等的编委。