On the Number of Affine Equivalence Classes of Boolean Functions

TL;DR

This paper derives explicit and asymptotic formulas for counting affine linear group orbits on Reed-Muller codes, addressing open questions and providing theoretical solutions to previously computed cases.

Contribution

It provides the first explicit formula for the number of AGL orbits of R(n,n) and an asymptotic formula for R(n,n)/R(1,n), solving open problems in the field.

Findings

Explicit formula for AGL orbits of R(n,n).

Asymptotic formula for AGL orbits of R(n,n)/R(1,n).

Answers to open questions by MacWilliams and Sloane.

Abstract

Let $R(r,n)$ be the $r$th order Reed-Muller code of length $2^n$. The affine linear group $\text{AGL}(n,\Bbb F_2)$ acts naturally on $R(r,n)$. We derive two formulas concerning the number of orbits of this action: (i) an explicit formula for the number of AGL orbits of $R(n,n)$, and (ii) an asymptotic formula for the number of AGL orbits of $R(n,n)/R(1,n)$. The number of AGL orbits of $R(n,n)$ has been numerically computed by several authors for $n\le 10$; result (i) is a theoretic solution to the question. Result (ii) answers a question by MacWilliams and Sloane.