Anbar, Nurdagül and Meidl, Wilfried and Topuzoğlu, Alev (2016) Idempotent and ppotent quadratic functions: distribution of nonlinearity and codimension. Designs, Codes and Cryptography . ISSN 09251022 (Print) 15737586 (Online) Published Online First http://dx.doi.org/10.1007/s1062301602138
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Official URL: http://dx.doi.org/10.1007/s1062301602138
Abstract
The Walsh transform QˆQ^ of a quadratic function Q:Fpn→FpQ:Fpn→Fp satisfies Qˆ(b)∈{0,pn+s2}Q^(b)∈{0,pn+s2} for all b∈Fpnb∈Fpn , where 0≤s≤n−10≤s≤n−1 is an integer depending on Q. In this article, we study the following three classes of quadratic functions of wide interest. The class C1C1 is defined for arbitrary n as C1={Q(x)=Trn(∑⌊(n−1)/2⌋i=1aix2i+1):ai∈F2}C1={Q(x)=Trn(∑i=1⌊(n−1)/2⌋aix2i+1):ai∈F2} , and the larger class C2C2 is defined for even n as C2={Q(x)=Trn(∑(n/2)−1i=1aix2i+1)+Trn/2(an/2x2n/2+1):ai∈F2}C2={Q(x)=Trn(∑i=1(n/2)−1aix2i+1)+Trn/2(an/2x2n/2+1):ai∈F2} . For an odd prime p, the subclass DD of all pary quadratic functions is defined as D={Q(x)=Trn(∑⌊n/2⌋i=0aixpi+1):ai∈Fp}D={Q(x)=Trn(∑i=0⌊n/2⌋aixpi+1):ai∈Fp} . We determine the generating function for the distribution of the parameter s for C1,C2C1,C2 and DD . As a consequence we completely describe the distribution of the nonlinearity for the rotation symmetric quadratic Boolean functions, and in the case p>2p>2 , the distribution of the codimension for the rotation symmetric quadratic pary functions, which have been attracting considerable attention recently. Our results also facilitate obtaining closed formulas for the number of such quadratic functions with prescribed s for small values of s, and hence extend earlier results on this topic. We also present the complete weight distribution of the subcodes of the second order Reed–Muller codes corresponding to C1C1 and C2C2 in terms of a generating function.
Item Type:  Article 

Uncontrolled Keywords:  Quadratic functions, Plateaued functions, Bent functions, Walsh transform, Idempotent functions, Rotation symmetric, ReedMuller code 
Subjects:  Q Science > QA Mathematics > QA150272.5 Algebra 
Divisions:  Faculty of Engineering and Natural Sciences > Basic Sciences > Mathematics Faculty of Engineering and Natural Sciences 
Depositing User:  Alev Topuzoğlu 
Date Deposited:  07 Nov 2016 12:05 
Last Modified:  09 Sep 2017 16:58 
URI:  https://research.sabanciuniv.edu/id/eprint/30811 
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Idempotent and ppotent quadratic functions: distribution of nonlinearity and codimension. (deposited 25 Dec 2015 14:37)
 Idempotent and ppotent quadratic functions: distribution of nonlinearity and codimension. (deposited 07 Nov 2016 12:05) [Currently Displayed]