Characters and cyclotomic fields in finite geometry / Bernhard Schmidt

This monograph contributes to the existence theory of combinatorial objects admitting certain types of automorphism groups. Some combinatorial objects such as difference sets, planar functions, group invariant weighing matrices, two-weight irreducible cyclic codes, can be studied in terms of group ring equations. Using a method free from severe technical assumptions, the author provides nonexistence theorems of broader applicability, making real progress in studying the circulant Hadamard matrix conjecture, Ryser’s conjecture and the Baker conjecture. This method is developed in the Chapter 2 and is the method of “field descent” which consists in the fact that cyclotomic integers X such that X 2 is rational usually are contained in a much smaller cyclotomic field as expected a priori. Other major contributions are an improvement of Turyn’s self-conjugacy exponent bound, by a severe refinement of Turyn’s method, and a classification of two-weight irreducible cyclic codes, by using Fourier analysis in abelian groups. In spite of the fact that some chapters can be read by people knowing basic algebra, the book is addressed to specialists. (Zentralblatt)