On the Number of Affine Equivalence Classes of Boolean Functions and q-Ary Functions

Let R
_q
(r,n) be the rth order q-ary Reed-Muller code of length q
ⁿ
, which is the set of functions from \mathbb F
_q
ⁿ
to \mathbb F
_q
represented by polynomials of degree ≤ r in \mathbb F
_q
[X
₁
, ... ,X
_n
]. The affine linear group AGL(n,\mathbb F
_q
) acts naturally on R
_q
(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
_q
(n(q-1),n), and (ii) an asymptotic formula for the number of AGL orbits of R
₂
(n,n)/R
₂
(1,n). The number of AGL orbits of R
₂
(n,n) has been numerically computed by several authors for n ≤ 31; the binary case of result (i) is a theoretic solution to the question. Result (ii) answers a question by MacWilliams and Sloane.