Idempotent and p-potent quadratic functions: distribution of nonlinearity and co-dimension

Abstract

The Walsh transform $$\widehat{Q}$$Q^ of a quadratic function $$Q:{\mathbb F}_{p^n}\rightarrow {\mathbb F}_p$$Q:FpnźFp satisfies $$|\widehat{Q}(b)| \in \{0,p^{\frac{n+s}{2}}\}$$|Q^(b)|ź{0,pn+s2} for all $$b\in {\mathbb F}_{p^n}$$bźFpn, where $$0\le s\le n-1$$0≤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 $$\mathcal {C}_1$$C1 is defined for arbitrary n as $$\mathcal {C}_1 = \{Q(x) = \mathrm{Tr_n}(\sum _{i=1}^{\lfloor (n-1)/2\rfloor }a_ix^{2^i+1})\;:\; a_i \in {\mathbb F}_2\}$$C1={Q(x)=Trn(źi=1ź(n-1)/2źaix2i+1):aiźF2}, and the larger class $$\mathcal {C}_2$$C2 is defined for even n as $$\mathcal {C}_2 = \{Q(x) = \mathrm{Tr_n}(\sum _{i=1}^{(n/2)-1}a_ix^{2^i+1}) + \mathrm{Tr_{n/2}}(a_{n/2}x^{2^{n/2}+1}) \;:\; a_i \in {\mathbb F}_2\}$$C2={Q(x)=Trn(źi=1(n/2)-1aix2i+1)+Trn/2(an/2x2n/2+1):aiźF2}. For an odd prime p, the subclass $$\mathcal {D}$$D of all p-ary quadratic functions is defined as $$\mathcal {D} = \{Q(x) = \mathrm{Tr_n}(\sum _{i=0}^{\lfloor n/2\rfloor }a_ix^{p^i+1})\;:\; a_i \in {\mathbb F}_p\}$$D={Q(x)=Trn(źi=0źn/2źaixpi+1):aiźFp}. We determine the generating function for the distribution of the parameter s for $$\mathcal {C}_1, \mathcal {C}_2$$C1,C2 and $$\mathcal {D}$$D. As a consequence we completely describe the distribution of the nonlinearity for the rotation symmetric quadratic Boolean functions, and in the case $$p > 2$$p>2, the distribution of the co-dimension for the rotation symmetric quadratic p-ary 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 $$\mathcal {C}_1$$C1 and $$\mathcal {C}_2$$C2 in terms of a generating function.