报告一：MSR codes with linear field size and smallest sub-packetization for any number of helper nodes (胡思煌)
报告时间：2024年10月31日15:00-17:00
报告地点：理科大楼B1202
报告摘要:
The sub-packetization $\ell$ and the field size $q$ are of paramount importance in the MSR array code constructions. For optimal-access MSR codes, Balaji \emph{et al.} proved that $\ell\geq s^{\left\lceil n/s \right\rceil}$, where $s = d-k+1$. Rawat \emph{et al.} showed that this lower bound is attainable for all admissible values of $d$ when the field size is exponential in $n$. After that, tremendous efforts have been devoted to reducing the field size. However, till now, reduction to linear field size is only available for $d\in\{k+1,k+2,k+3\}$ and $d=n-1$.
In this work, we construct the first class of explicit optimal-access MSR codes with the smallest sub-packetization $\ell = s^{\left\lceil n/s \right\rceil}$ for all $d$ between $k+1$ and $n-1$, resolving an open problem in the survey (Ramkumar \emph{et al.}, Foundations and Trends in Communications and Information Theory: Vol. 19: No. 4). We further propose another class of explicit MSR code constructions (not optimal-access) with even smaller sub-packetization $s^{\left\lceil n/(s+1)\right\rceil }$ for all admissible values of $d$, making significant progress on another open problem in the survey. Previously, MSR codes with $\ell=s^{\left\lceil n/(s+1)\right\rceil }$ and $q=O(n)$ were only known for $d=k+1$ and $d=n-1$. The key insight that enables a linear field size in our construction is to reduce $\binom{n}{r}$ global constraints of non-vanishing determinants to $O_s(n)$ local ones, which is achieved by carefully designing the parity check matrices. This is a joint work with Guodong Li, Ningning Wang, and Min Ye.
报告人简介:
胡思煌，山东大学网络空间安全学院教授，目前主要研究方向是通信与存储编码理论。在组合数学与信息论期刊和会议上发表30余篇论文，主持国家重点研发计划青年科学家项目、基金委青年项目和CCF-华为胡杨林基金，指导博士生获2024 IEEE Jack Keil Wolf ISIT Student Paper Award。
报告二：Deep Holes of Twisted Reed-Solomon Codes （方伟军）
报告时间：2024年10月31日15:00-17:00
报告地点：理科大楼B1202
报告摘要:
The deep holes of a linear code are the vectors that achieve the maximum error distance to the code. There has been extensive research on the topic of deep holes in Reed-Solomon codes. In this talk, as a generalization of Reed-Solomon codes, we investigate the problem of deep holes in a class of twisted Reed-Solomon codes. The covering radius and a standard class of deep holes of twisted Reed-Solomon codes are obtained for a general evaluation set. Furthermore, we completely determine the deep holes of the full-length twisted Reed-Solomon codes with parameters in a certain range.
报告人简介:
方伟军，山东大学网络空间安全学院教授、博士生导师。2019年博士毕业于南开大学，导师为符方伟教授。 2019-2021年在清华大学从事博士后工作，导师为夏树涛教授。目前主要研究方向为代数编码、量子纠错码以及分布式存储编码等。在IEEE TIT、FFA、DCC以及ISIT等信息论与编码理论主流期刊与会议发表二十余篇论文。主持1项国家自然科学青年基金，作为学术骨干参与国家重点研发计划项目和青年科学家项目。获得山东大学齐鲁青年学者，入选山东省泰山学者青年专家人才计划。