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Asymptotics of Moore exponent sets
日期:2020/10/12 12:38:06  发布:数学科学学院  浏览: 1154

时间:2020年10月17日(星期六)  14:30-15:30

地点:腾讯会议 ID:570 992 730


报告题目:Asymptotics of Moore exponent sets
报告人:周悦

    报告人简介:周悦博士现任国防科技大学数学系副研究员,曾获德国“洪堡”学者资助,2016年度Kirkman奖章获得者。在Adv. Math, JCTA等著名期刊上发表SCI论文近30篇,主持国家自然科学基金面上项目1项,青年项目1项,获湖南省优秀青年基金资助。

    报告摘要:Let $n$ be a positive integer and $I$ a $k$-subset of integers in $[0,n-1]$. Given a $k$-tuple $A=(\alpha_0, \cdots, \alpha_{k-1})\in \F^k_{q^n}$, let $M_{A,I}$ denote the matrix $(\alpha_i^{q^j})$ with $0\leq i\leq k-1$ and $j\in I$. When $I=\{0,1,\cdots, k-1\}$, $M_{A,I}$ is called a Moore matrix which was introduced by E.\ H.\ Moore in 1896. It is well known that the determinant of a Moore matrix equals $0$ if and only if $\alpha_0,\cdots, \alpha_{k-1}$ are $\F_q$-linearly dependent. We call $I$ that satisfies this property a \emph{Moore exponent set}. In fact, Moore exponent sets are equivalent to maximum rank-distance (MRD) code with maximum left and right idealisers over finite fields. It is already known that $I=\{0,\cdots, k-1\}$ is not the unique Moore exponent set, for instance, (generalized) Delsarte-Gabidulin codes and the MRD codes recently discovered in \cite{csajbok_MRD_maximum_2020} both give rise to new Moore exponent sets.

    By using algebraic geometry approach, we obtain an asymptotic classification result: for $q>5$ and given $I$, there exists an integer $N$ depending on $\max\{i: i\in I\}$ such that if $n>N$ and $I$ is a Moore exponent set then $I$ must be an arithmetic progression.

    This talk is based on two recent joint works \cite{bartoli_asymptotoics_2020} with Daniele Bartoli and \cite{csajbok_MRD_maximum_2020} with Bence Csajb{\'o}k, Giuseppe Marino and Olga Polverino.

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