Some New Cyclotomic Strongly Regular Graphs

Abstract

Given a finite field F and a subset D of F* such that D = —D, we can define a graph F with vertex set F by letting x ~ y whenever y x 6 D. (Here ~ denotes adjacency.) The spectrum of F consists of the numbers £deD X (d), where x runs through the (additive) characters of F. In particular, the trivial character x0 yields the eigenvalue | D \, the valency of F. One might wonder in what cases the graph F is strongly regular, and there has been done a lot of work on this question, see, e.g., Delsarte [4], van Lint and Schrijver [6], Calderbank and Kantor [3], Brouwer [1], de Resmini [7] and de Resmini and Migliori [8]. (As Delsarte showed, there is a one-to-one correspondence between (i) sets D closed under multiplication by elements of the prime field of F (and yielding a strongly regular F), and (ii) projective two-weight codes, and (iii) subsets of projective spaces such that the cardinality of the intersection with a hyperplane takes only two values. Work on this problem occurs in each of these three terminologies.) In [5], we constructed four new examples, which will be described below. Our sets D will be unions of a number of cosets of a subgroup K of F*, i.e., D = Z K for some set Z c F*. The field F is described by its characteristic p and a primitive polynomial defining it over its prime field. For the resulting strongly regular graphs we give the standard parameters v, k, A, m, r, s, f, g (cf. Brouwer and van Lint [2]).