On the Maiorana-McFarland Class Extensions

The closure ${\mathcal{M}}_m^{\mathrm{\#}}$ 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 $f\oplus {\mathrm{I}\mathrm{n}\mathrm{d}}_U$ are considered, where $U$ is an affine subspace of ${\mathbb{F}}_2^m$ of dimension $m\mathrm{/}2$. We obtain an explicit formula for $\mathrm{\mid }{\widehat{\mathcal{M}}}_m\mathrm{\mid }$ and an upper bound for $\mathrm{\mid }{\widehat{\mathcal{M}}}_m^{\mathrm{\#}}\mathrm{\mid }$. Asymptotically tight bounds for $\mathrm{\mid }{\mathcal{M}}_m^{\mathrm{\#}}\mathrm{\mid }$ are proved as well, for instance, $\mathrm{\mid }{\mathcal{M}}_8^{\mathrm{\#}}\mathrm{\mid }\approx 2^{77.865}$. Metric properties of ${\mathcal{M}}_m$ and ${\mathcal{M}}_m^{\mathrm{\#}}$ 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 ${\mathcal{M}}_m^{\mathrm{\#}}$. The average number $E({\mathcal{M}}_m)$ of $m\mathrm{/}2$-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 $E({\mathcal{M}}_m^{\mathrm{\#}})$ satisfies $E({\mathcal{M}}_m^{\mathrm{\#}})<E({\mathcal{M}}_m)$ and $E({\mathcal{M}}_m^{\mathrm{\#}})=E({\mathcal{M}}_m)-o(1)$.