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

The Walsh transform $\widehat{Q}$ of a quadratic function $Q:F_{p^n}\rightarrow F_p$ satisfies $|\widehat{Q}(b)| \in \{0,p^{\frac{n+s}{2}}\}$ for all $b\in F_{p^n}$, 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 $\mathcal{C}_1 = \{Q(x) = Tr(\sum_{i=1}^{\lfloor (n-1)/2\rfloor}a_ix^{2^i+1})\;:\; a_i \in F_2\}$, and the larger class $\mathcal{C}_2$ is defined for even $n$ as $\mathcal{C}_2 = \{Q(x) = Tr(\sum_{i=1}^{(n/2)-1}a_ix^{2^i+1}) + {\rm Tr_{n/2}}(a_{n/2}x^{2^{n/2}+1}) \;:\; a_i \in F_2\}$. For an odd prime $p$, the subclass $\mathcal{D}$ of all $p$-ary quadratic functions is defined as $\mathcal{D} = \{Q(x) = Tr(\sum_{i=0}^{\lfloor n/2\rfloor}a_ix^{p^i+1})\;:\; a_i \in F_p\}$. 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 $\mathcal{C}_2$.