7 - Cyclotomic constructions

Summary

In this chapter, we look at graphs defined by a difference set in a usually abelian group. Difference sets in a vector space that are invariant under multiplication by scalars are equivalent to two-weight codes and to two-character subsets of a projective space. We survey a lot of examples of such two-character sets (infinite families and sporadic ones, the latter summarised in a table). We review cyclic codes, in particular cyclic two-weight codes and introduce the related Van Lint-Schrijver graphs, the Hill graph, the De Lange graphs and the Peisert graphs. Then our attention goes to the one-dimensional affine rank 3 graphs, which we review in some detail, including proofs of the parameter restrictions that lead to the different cases: the Paley graphs, the Van Lint-Schrijver graphs and the Peisert graphs. We also discuss the Paley graphs in some detail and provide a table with small strongly regular power residue graphs. The penultimate section is dedicated to graphs related to the action of the alternating group Alt(5) and the symmetric group Sym(4) on a projective line. In the last section, we review strongly regular graphs constructed from bent functions.