新型代数编码及其应用研讨会会议安排

四川 成都 新葡萄8883官网AMG2023年10月29日-2023年10月31日

本次会议主题包括(但不限于)：有限域及其相关数学理论在代数编码领域的应用。旨在为相关学者提供一个学术平台，交流最新发展动态及学术成果，促进有限域、代数编码理论等相关领域的交叉、融合与发展，为该领域的老师、学生提供一个相互学习和交流的场所。

联系人：阎昊德，罗荣  邮箱：hdyan@swjtu.edu.cn luorong@swjtu.edu.cn

报告摘要

报告题目: Ractional Revival on Abelian Cayley Graphs

报告人：曹喜望教授

摘要: Fractional revival, known as a quantum transport phenomenon, is essential for entanglement generation in quantum spin networks. The concept of fractional revival is a generalization of perfect state transfer and periodicity on graphs. In this talk, we propose a sufficient and necessary condition for abelian Cayley graphs having fractional revival between any two distinct vertices. With this characterization, two general constructions of abelian Cayley graphs having fractional revival is presented. Meanwhile, we establish several new families of abelian Cayley graphs admitting fractional revival. This is a joint work with Gaojun Luo.

报告题目：Constacyclic MDS Codes and GRS Codes

报告人：刘宏伟教授

摘要：Maximum distance separable (MDS) codes are optimal in the sense that the minimum distance cannot be improved for a given length and code size. The most prominent MDS codes are generalized Reed–Solomon (GRS) codes. The square C^2 of a linear code C is the linear code spanned by the component-wise products of every pair of codewords in C. For an MDS code C, it is convenient to determine whether C is a GRS code by determining the dimension of C^2. In this talk, we investigate under what conditions that constacyclic MDS codes are GRS. For this purpose, we first study the square of constacyclic codes. We then give a sufficient condition that a constacyclic code is GRS. In particular, we provide a necessary and sufficient condition that a constacyclic code of a prime length is GRS. This talk is based on joint work with Shengwei Liu (Designs, Codes and Cryptography, https://doi.org/10.1007/s10623-023-01294-6, 2023.)

报告题目：More Differential Properties about the Ness-Helleseth Nonlinear Mapping

报告人：夏永波教授

摘要：Let n be an odd integer, d_1=(3^n-1)/2-1 and d_2=3^n-2. The function defined by f_u (x)=(ux)^d1+x^d2  is called the Ness-Helleseth mapping from F_(3^n ) to itself, where u in F_(3^n ) . In this talk, we show that f_u (x) is an almost perfect nonlinear (APN) mapping if and only if χ(u + 1) = χ(u - 1) = χ(u), where χ(·) denotes the quadratic character of F_(3^n ). This settles the open problem raised by Ness and Helleseth in IEEE Trans Inform Theory 53(7): 2581-2586, 2007, where they only gave the proof of sufficiency for the above result. In addition, the differential properties of f_u (x) are further investigated. For each u in F_3   or u satisfying χ(u+1) = χ(u-1), the differential spectrum of f_u (x) is determined, and in some cases it is expressed in terms of quadratic character sums involving cubic polynomials. Our results can also be generalized to the case p ≡ 3 (mod 4).

报告题目：On non-expandable cross-bifix-free codes

报告人：陈博聪教授

报告摘要：A cross-bifix-free code of length n over Z_q is defined as a non-empty subset of Z_q^n satisfying that the prefix set of each codeword is disjoint from the suffix set of every codeword. Cross-bifix-free codes have found important applications in digital communication systems. One of the main research problems on cross-bifix-free codes is to construct cross-bifix-free codes as large as possible in size. Recently, Wang and Wang introduced a family of cross-bifix-free codes S_(I,J)^((k)) (n), which is a generalization of the classical cross-bifix-free codes studied early by Lvenshtein, Gilbert and Chee et al.. It is known that S_(I,J)^k (n) is nearly optimal in size and S_(I,J)^k (n) is non-expandable if k = n − 1 or 1 ≤ k < n/2. In this talk, we first show that S_(I,J)^((k)) (n) is non-expandable if and only if k = n − 1 or 1 ≤ k < n/2, thereby improving the results in [Chee et al., IEEE-TIT, 2013] and [Wang and Wang, IEEE-TIT, 2022]. We then construct a new family of cross-bifix-free codes U_(I,J)^((k)) (n) to expand S_(I,J)^((k)) (n) such that the resulting larger code S_(I,J)^((k)) (n)∪U_(I,J)^((k)) (n) is a non-expandable cross-bifix-free code whenever S_(I,J)^((k)) (n) is expandable. Finally, we present an explicit formula for the size of S_(I,J)^((k)) (n)∪U_(I,J)^((k)) (n). This talk is based on a joint work with Chunyan Qin and Gaojun Luo.