An asymptotic lower bound on the number of bent functions

TL;DR

This paper establishes a new asymptotic lower bound on the number of bent Boolean functions, leveraging modifications of known function families and recent combinatorial estimates.

Contribution

It introduces a novel asymptotic lower bound on the count of bent functions using modifications of the Maiorana--McFarland family and combinatorial estimations.

Findings

Derived asymptotics for the number of bent functions.

Connected bent function enumeration to Latin square transversals.

Provided asymptotic counts for hypercube partitions.

Abstract

A Boolean function $f$ on $n$ variables is said to be a bent function if the absolute value of all its Walsh coefficients is $2^{n/2}$. Our main result is a new asymptotic lower bound on the number of Boolean bent functions. It is based on a modification of the Maiorana--McFarland family of bent functions and recent progress in the estimation of the number of transversals in latin squares and hypercubes. By-products of our proofs are the asymptotics of the logarithm of the numbers of partitions of the Boolean hypercube into $2$-dimensional affine and linear subspaces.