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arXiv · 2007.12308

On the Number of Affine Equivalence Classes of Boolean Functions

Xiang-dong Hou
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Abstract

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.

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Xiang-dong Hou. 2021-08-18. On the Number of Affine Equivalence Classes of Boolean Functions. https://arxiv.org/abs/2007.12308

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