任意旋转对称布尔函数的权递归
Weight recursions for any rotation symmetric Boolean functions
摘要：
令$f_n(x_1, x_2, \ldots, x_n)$表示次数为$d$的旋转对称布尔函数在$n \geq d$变量中的代数范式（多项式形式），并且令$wt(f_n)$表示该函数的汉明重量。令$(1, a_2, \ldots, a_d)_n$表示由单式$x_1x_{a_2} \cdots x_{a_d}.$生成的$n$变量中的次数$d$的函数$f_n$。这样的函数$f_n$被称为{ \em单式旋转对称}（MRS）。在$2012$论文中证明了对于任何具有$d=3,$的MRS $f_n$，权序列$\{w_k = wt(f_k):~k = 3, 4, \ldots\}$满足具有整数系数的齐次线性递归。本文证明了对于任何旋转对称函数$f_n;$，都存在这样的递归式，这样的函数是由$t$各次单项式的某个和生成的。本文的最后一节给出了一个Mathematica程序，它明确地计算齐次线性递归的权重，给定任何旋转对称$f_n.$读者谁是唯一感兴趣的是找到一些递归可以使用该程序，而不是关心的细节，而复杂的证明在本文中。
英文摘要：
Let $f_n(x_1, x_2, \ldots, x_n)$ denote the algebraic normal form (polynomial form) of a rotation symmetric Boolean function of degree $d$ in $n \geq d$ variables and let $wt(f_n)$ denote the Hamming weight of this function. Let $(1, a_2, \ldots, a_d)_n$ denote the function $f_n$ of degree $d$ in $n$ variables generated by the monomial $x_1x_{a_2} \cdots x_{a_d}.$ Such a function $f_n$ is called {\em monomial rotation symmetric} (MRS). It was proved in a $2012$ paper that for any MRS $f_n$ with $d=3,$ the sequence of weights $\{w_k = wt(f_k):~k = 3, 4, \ldots\}$ satisfies a homogeneous linear recursion with integer coefficients. In this paper it is proved that such recursions exist for any rotation symmetric function $f_n;$ such a function is generated by some sum of $t$ monomials of various degrees. The last section of the paper gives a Mathematica program which explicitly computes the homogeneous linear recursion for the weights, given any rotation symmetric $f_n.$ The reader who is only interested in finding some recursions can use the program and not be concerned with the details of the rather complicated proofs in this paper.
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