Spectra of a class of quadratic functions: Average behaviour and counting functions

The Walsh transform Q̂$\widehat {Q}$ of a quadratic function Q:Fpn→Fp$Q:\mathbb {F}_{p^{n}}\rightarrow \mathbb {F}_{p}$ satisfies |Q̂|∈{0,pn+s2}$|\widehat {Q}| \in \{0,p^{\frac {n+s}{2}}\}$ for an integer 0 ≤ s ≤ n−1. We study quadratic functions given in trace form Q(x)=Trn(∑i=0kaixpi+1)$Q(x) = {{\text {Tr}_{\mathrm {n}}}}({\sum }_{i=0}^{k}a_{i}x^{p^{i}+1})$ with the restriction that ai∈Fp,0≤i≤k$a_{i} \in \mathbb {F}_{p},~ 0\leq i\leq k$. We determine the expected value for the parameter s for such quadratic functions from Fpn$\mathbb {F}_{p^{n}}$ to Fp$\mathbb {F}_{p}$, for many classes of integers n. Our exact formulas confirm that on average the value of s is small, and hence the average nonlinearity of this class of quadratic functions is high when p = 2. We heavily use methods, recently developed by Meidl, Topuzoğlu and Meidl, Roy, Topuzoğlu in order to construct/enumerate such functions with prescribed s. In the first part of this paper we describe these methods in detail and summarize the counting results.