Quadratic functions with prescribed spectra

We study a class of quadratic p-ary functions $${{\mathcal{F}}_{p,n}}$$ from $${\mathbb{F}_{p^n}}$$ to $${\mathbb{F}_p, p \geq 2}$$, which are well-known to have plateaued Walsh spectrum; i.e., for each $${b \in \mathbb{F}_{p^n}}$$ the Walsh transform $${\hat{f}(b)}$$ satisfies $${|\hat{f}(b)|^2 \in \{ 0, p^{(n+s)}\}}$$ for some integer 0 ≤ s ≤ n − 1. For various types of integers n, we determine possible values of s, construct $${{\mathcal{F}}_{p,n}}$$ with prescribed spectrum, and present enumeration results. Our work generalizes some of the earlier results, in characteristic two, of Khoo et. al. (Des Codes Cryptogr, 38, 279–295, 2006) and Charpin et al. (IEEE Trans Inf Theory 51, 4286–4298, 2005) on semi-bent functions, and of Fitzgerald (Finite Fields Appl 15, 69–81, 2009) on quadratic forms.