2016/03/15 by Nurdagül Anbar, Wilfried Meidl, Anbar, Nurdagül +4
Computer Science · Mathematics · #Analytic Number Theory Research #Analytic and geometric function theory #Binary quadratic form #Coding theory and cryptography #Combinatorics #Dimension (graph theory) #Discrete mathematics #Distribution (mathematics) #FOS: Mathematics #Finite Group Theory Research #Geometry #Idempotence #Integer (computer science) #Mathematical analysis #Mathematics #Number Theory (math.NT) #Quadratic equation #Quadratic form (statistics) #Quadratic function #math.NT #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1603.04685
published in arXiv (Cornell University) (Cornell University)
arxiv created 2016/03/15 · openalex publication_date 2016/03/15 · arxiv updated 2016/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
The Walsh transform widehatQ of a quadratic function\nQ:Fpn\→ Fp satisfies | widehatQ(b)| \∈\n 0,p\(n+s)/(2) for all b\∈ Fpn, where 0\≤ s\≤ n-1 is an\ninteger depending on Q. In this article, we study the following three classes\nof quadratic functions of wide interest. The class \C1 is defined\nfor arbitrary n as \C1 = Q(x) = Tr(\∑i=1 lfloor\n(n-1)/2 rflooraix2i+1) ;: ; ai \∈ F2 , and the larger class\n\C2 is defined for even n as \C2 = Q(x) =\nTr(\∑i=1(n/2)-1aix2i+1) + rm Trn/2(an/2x^2n/2+1)\n ;: ; ai \∈ F2 . For an odd prime p, the subclass \D of all\np-ary quadratic functions is defined as \D = Q(x) =\nTr(\∑i=0 lfloor n/2 rflooraixpi+1) ;: ; ai \∈ Fp . We\ndetermine the distribution of the parameter s for \C1,\n\C2 and \D. As a consequence we obtain the distribution\nof the nonlinearity for the rotation symmetric quadratic Boolean functions, and\nin the case p > 2, our results yield the distribution of the co-dimensions\nfor the rotation symmetric quadratic p-ary functions, which have been\nattracting considerable attention recently. We also present the complete weight\ndistribution of the subcodes of the second order Reed-Muller codes\ncorresponding to \C1 and \C2.\n