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

Let {R}_{q}({r},{n}) be the {r} th 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 \le {r} in {\mathbb {F}}_{q}[{X}_{1}, {\dots },{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}\le 31 ; the binary case of result (i) is a theoretic solution to the question. Result (ii) answers a question by MacWilliams and Sloane.