Combinatorial t-designs from special polynomials

2019/03/18 by Ding, Cunsheng, Tang, Chunming
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1903.07375

Abstract

Combinatorial t-designs have nice applications in coding theory, finite geometries and several engineering areas. There are two major methods of constructing t-designs. One of them is via group actions of certain permutation groups which are t-transitive or t-homogeneous on some point set. The other is a coding-theoretical one. The objectives of this paper are to introduce two constructions of t-designs with special polynomials over finite fields GF(q), and obtain 2-designs and 3-designs with interesting parameters. A type of d-polynomials is defined and used to construct 2-designs. Under the framework of the first construction, it is shown that every o-polynomial over GF(2^m) gives a 2-design, and every o-monomial over GF(2^m) yields a 3-design. Under the second construction, every o-polynomial gives a 3-design. Some open problems and conjectures are also presented in this paper.

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