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An asymptotic lower bound on the number of bent functions

V. N. Potapov, A. A. Taranenko, Yu. V. Tarannikov

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.

An asymptotic lower bound on the number of bent functions

Abstract

A Boolean function on variables is said to be a bent function if the absolute value of all its Walsh coefficients is . 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 -dimensional affine and linear subspaces.

Paper Structure

This paper contains 4 sections, 14 theorems, 48 equations.

Table of Contents

  1. Definitions, preliminaries, and main results
  2. Construction of bent functions
  3. Proof of the lower bound
  4. Bounds on the number of $2$-spreads

Key Result

Theorem 1

Let $b_n$ be the number of bent functions on $n$ variables. Then it is not greater than $6^{3\cdot2^{n-6}}2^{\cdot2^{n-2}(1+o(1))}$ as $n\rightarrow\infty$. In particular,

Theorems & Definitions (22)

  • Theorem 1: potapov.upbent
  • Theorem 2
  • Proposition 1
  • Proposition 2: taran.specplat
  • proof
  • Theorem 3: BakTar.onebentfunc
  • proof
  • Theorem 4
  • Theorem 5
  • Proposition 3
  • ...and 12 more