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

The Walsh transform $\widehat{Q}$ of a quadratic function $Q:F_{p^n}\to F_p$ satisfies $|\widehat{Q}(b)| \in {0,p^{\frac{n+s}{2}}}$forall$b\in F_{p^n}$,where$0\le s\le n-1$ is an integer depending on $Q$. In this article, we study the following three classes of quadratic functions of wide interest. The class ${\mathcal{C}}_1$ is defined for arbitrary $n$ as $$\mathcal{C}$1 = {Q(x) = Tr(\sum{i=1}^{\lfloor (n-1)/2\rfloor}a_ix^{2^i+1});:; a_i \in F_2}$,andthelargerclass$$\mathcal{C}$2$isdefinedforeven$n$as$$\mathcal{C}$2 = {Q(x) = Tr(\sum{i=1}^{(n/2)-1}a_ix^{2^i+1}) + {\rm Tr{n/2}}(a_{n/2}x^{2^{n/2}+1}) ;:; a_i \in F_2}$\mathrm{.}Foranoddprime$p$,thesubclass$$\mathcal{D}$$ofall$p$-aryquadraticfunctionsisdefinedas$$\mathcal{D}$ = {Q(x) = Tr(\sum_{i=0}^{\lfloor n/2\rfloor}a_ix^{p^i+1});:; a_i \in F_p}$. We determine the distribution of the parameter $s$ for $$\mathcal{C}$_1, $\mathcal{C}$_2$and$$\mathcal{D}$$. As a consequence we obtain the distribution of the nonlinearity for the rotation symmetric quadratic Boolean functions, and in the case $p>2$, our results yield the distribution of the co-dimensions for the rotation symmetric quadratic $p$-ary functions, which have been attracting considerable attention recently. We also present the complete weight distribution of the subcodes of the second order Reed-Muller codes corresponding to ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$.