On the Number of Affine Equivalence Classes of Boolean Functions

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.