Weight recursions for any rotation symmetric Boolean functions

TL;DR

This paper proves that the weights of any rotation symmetric Boolean function follow a homogeneous linear recursion and provides a Mathematica program to compute these recursions explicitly.

Contribution

It generalizes previous results by establishing linear recursions for weights of all rotation symmetric Boolean functions, not just degree 3 cases.

Findings

Weights satisfy a homogeneous linear recursion for any rotation symmetric Boolean function.

A Mathematica program is provided to compute the recursions explicitly.

Recursions can be derived for functions generated by sums of monomials of various degrees.

Abstract

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.