Cyclotomy, difference sets, sequences with low correlation, strongly regular graphs, and related geometric substructures

Abstract

In this paper, we survey constructions of and nonexistence results on combinatorial/geometric structures which arise from unions of cyclotomic classes of finite fields. In particular, we survey both classical and recent results on difference sets related to cyclotomy, and cyclotomic constructions of sequences with low correlation. We also give an extensive survey of recent results on constructions of strongly regular Cayley graphs and related geometric substructures such as $m$-ovoids and $i$-tight sets in classical polar spaces.

29 pages. To appear in the Proceedings of the Workshop "Pseudorandomness and Finite Fields" at RICAM, Linz, Austria, Oct. 2018, The Radon Series on Computational and Applied Mathematics of de Gruyter, 2018/2019