论文标题
关于布尔功能的仿射等效类别的数量
On the Number of Affine Equivalence Classes of Boolean Functions
论文作者
论文摘要
令$ r（r，n）$为$ r $ th订单reed-muller长度$ 2^n $的代码。仿射线性组$ \ text {agl}（n，\ bbb f_2）$自然作用于$ r（r，n）$。我们得出了有关此操作轨道数量的两个公式：（i）$ r（n，n）$的AGL轨道数量的明确公式，以及（ii）$ r（n，n）/r（n）/r（1，n）$的agl轨道数量的渐近公式。 $ r（n，n）$的AGL轨道数量已由几位作者计算为$ n \ le 10 $；结果（i）是问题的理论解决方案。结果（ii）回答了麦克维利亚姆斯和斯隆的问题。
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.
