New cyclic difference sets with Singer parameters

Abstract

The main result in this paper is a general construction of φ(m)/2 pairwise inequivalent cyclic difference sets with Singer parameters (v,k,λ)=(2m-1,2m-1,2m-2) for any m≥3. The construction was conjectured by the second author at Oberwolfach in 1998. We also give a complete proof of related conjectures made by No, Chung and Yun and by No, Golomb, Gong, Lee and Gaal which produce another difference set for each m≥7 not a multiple of 3. Our proofs exploit Fourier analysis on the additive group of GF(2m) and draw heavily on the theory of quadratic forms in characteristic 2. By-products of our results are a new class of bent functions and a new short proof of the exceptionality of the Müller-Cohen-Matthews polynomials. Furthermore, following the results of this paper, there are today no sporadic examples of difference sets with these parameters; i.e. every known such difference set belongs to a series given by a constructive theorem. © 2003 Elsevier Inc. All rights reserved.