On the Maiorana-McFarland Class Extensions

The closure $\mathcal{M}_{m}^{\#}$ and the extension $\widehat{\mathcal{M}}_{m}$ of the Maiorana--McFarland class $\mathcal{M}_{m}$ in $m = 2n$ variables relative to the extended-affine equivalence and the bent function construction $f \oplus \mathrm{Ind}_{U}$ are considered, where $U$ is an affine subspace of $\mathbb{F}_{2}^{m}$ of dimension $m/2$. We obtain an explicit formula for $|\widehat{\mathcal{M}}_{m}|$ and an upper bound for $|\widehat{\mathcal{M}}_{m}^{\#}|$. Asymptotically tight bounds for $|\mathcal{M}_{m}^{\#}|$ are proved as well, for instance, $|\mathcal{M}_{8}^{\#}| \approx 2^{77.865}$. Metric properties of $\mathcal{M}_{m}$ and $\mathcal{M}_{m}^{\#}$ are also investigated. We find the number of all closest bent functions to the set $\mathcal{M}_{m}$ and provide an upper bound of the same number for $\mathcal{M}_{m}^{\#}$. The average number $E(\mathcal{M}_{m})$ of $m/2$-dimensional affine subspaces of $\mathbb{F}_{2}^{m}$ such that a function from $\mathcal{M}_{m}$ is affine on each of them is calculated. We obtain that similarly defined $E(\mathcal{M}_{m}^{\#})$ satisfies $E(\mathcal{M}_{m}^{\#}) < E(\mathcal{M}_{m})$ and $E(\mathcal{M}_{m}^{\#}) = E(\mathcal{M}_{m}) - o(1)$.