New cyclic difference sets with Singer parameters

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• 作者：Dillon ; J.F. ; Dobbertin ; Hans
• 关键词：Cyclic difference set ; Singer parameters ; Ideal autocorrelation ; Bent function ; Permutation polynomial ; Exceptional polynomial ; Dickson polynomial ; Mü ; ller&ndash ; Cohen&ndash ; Matthews polynomial
• 刊名：Finite Fields and Their Applications
• 出版年：2004
• 期刊代码：66_10715797
• 类别：mt
• 出版时间：July, 2004
• 卷：10
• 期：3
• 页码：342-389
• 文件大小：479 K

摘要

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)=(2^m−1,2^m−1,2^m−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(2^m) 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.