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

Abstract

Let <inline-formula> <tex-math notation= LaTeX >${R}_{q}({r},{n})$ </tex-math></inline-formula> be the <inline-formula> <tex-math notation= LaTeX >${r}$ </tex-math></inline-formula>th order <inline-formula> <tex-math notation= LaTeX >${q}$ </tex-math></inline-formula>-ary Reed-Muller code of length <inline-formula> <tex-math notation= LaTeX >${q}^{n}$ </tex-math></inline-formula>, which is the set of functions from <inline-formula> <tex-math notation= LaTeX >${\\mathbb {F}}_{q}^{n}$ </tex-math></inline-formula> to <inline-formula> <tex-math notation= LaTeX >${\\mathbb {F}}_{q}$ </tex-math></inline-formula> represented by polynomials of degree <inline-formula> <tex-math notation= LaTeX >$\\le {r}$ </tex-math></inline-formula> in <inline-formula> <tex-math notation= LaTeX >${\\mathbb {F}}_{q}[{X}_{1}, {\\dots },{X}_{n}]$ </tex-math></inline-formula>. The affine linear group <inline-formula> <tex-math notation= LaTeX >$AGL({n},{\\mathbb {F}}_{q})$ </tex-math></inline-formula> acts naturally on <inline-formula> <tex-math notation= LaTeX >${R}_{q}({r},{n})$ </tex-math></inline-formula>. We derive two formulas concerning the number of orbits of this action: (i) an explicit formula for the number of AGL orbits of <inline-formula> <tex-math notation= LaTeX >${R}_{q}({n}({q}-1),{n})$ </tex-math></inline-formula>, and (ii) an asymptotic formula for the number of AGL orbits of <inline-formula> <tex-math notation= LaTeX >${R}_{2}({n},{n})/{R}_{2}(1,{n})$ </tex-math></inline-formula>. The number of AGL orbits of <inline-formula> <tex-math notation= LaTeX >${R}_{2}({n},{n})$ </tex-math></inline-formula> has been numerically computed by several authors for <inline-formula> <tex-math notation= LaTeX >${n}\\le 31$ </tex-math></inline-formula>; the binary case of result (i) is a theoretic solution to the question. Result (ii) answers a question by MacWilliams and Sloane.