学术报告：Orthogonal arrays, dual codes, and divisibilities of polynomials over finite fields

题目: Orthogonal arrays, dual codes, and divisibilities of polynomials over finite fields

演讲人: Qiang Wang （Professor, Carleton University)

时间: 10:30-11:30, June 26, 2012

地点：信息工程研究所3号楼3221会议室

摘要: Consider a maximum-length shift-register sequence generated by a primitive polynomial $f$ over a finite field. The set of its subintervals is a linear code whose dual code is formed by all polynomials divisible by $f$. Since the minimum weight of dual codes is directly related to the strength of the corresponding orthogonal arrays, we can produce orthogonal arrays by studying divisibility of polynomials. Munemasa (Finite Fields Appl., 4(3):252-260, 1998) uses trinomials over $\mathbb{F}_2$  to construct orthogonal arrays of guaranteed strength $2$ (and almost strength 3). That result was extended by Dewar, Moura, Panario, Stevens and Wang (Des. Codes Cryptogr., 45:1-17, 2007) to construct orthogonal arrays of guaranteed strength $3$ by considering divisibility of trinomials by pentanomials over $\mathbb{F}_2$.

In this talk we review the above results and then extend some of the above results by dropping either the primitivity restriction or the binary field restriction. We also briefly comment on the combinatorial applications.   In particular, we show how we can use linear algebraic techniques (i.e., system of linear equations) to obtain some of our results.  This is a joint work with Daniel Panario, Olga Sosnovski, and Brett Stevens.