Maximally nonlinear functions over finite fields

An $n$-place function over a field $\mathbf {F}_q$ with $q$ elements is called maximally nonlinear if it has the largest nonlinearity among all $q$-valued $n$-place functions. We show that, for even $n \ge 2$, a function is maximally nonlinear if and only if its nonlinearity is $q^{n-1}(q - 1) - q^{\frac n2-1}$; for $n=1$, the corresponding criterion for maximal nonlinearity is $q-2$. For $q>2$ and even $n \ge 2$, we describe the set of all maximally nonlinear quadratic functions and find its cardinality. In this case, all maximally nonlinear quadratic functions are quadratic bent functions and their number is smaller than the halved number of the bent functions.