New cyclic difference sets with Singer parameters

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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.

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Dillon, J. F., & Dobbertin, H. (2004). New cyclic difference sets with Singer parameters. Finite Fields and Their Applications, 10(3), 342–389. https://doi.org/10.1016/j.ffa.2003.09.003

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