On the Number of Affine Equivalence Classes of Boolean Functions and q-Ary Functions
Xiang‐dong Hou · IEEE Transactions on Information Theory · 2021
Let Rq(r,n) be the rth order q-ary Reed-Muller code of length qn, which is the set of functions from \mathbb Fqnto \mathbb Fqrepresented by polynomials of degree ≤ r in \mathbb Fq[X1, ... ,Xn]. The affine linear group AGL(n,\mathbb Fq) acts naturally on Rq(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 Rq(n(q-1),n), and (ii) an asymptotic formula for the number of AGL orbits of R2(n,n)/R2(1,n). The number of AGL orbits of R2(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.