Recursions for quadratic rotation symmetric functions weights

TL;DR

This paper proves the Easy Coefficients Conjecture for quadratic rotation symmetric Boolean functions when certain conditions are met, simplifying weight calculations and enabling efficient computation of generating functions.

Contribution

It establishes the ECC under specific rules matrix conditions, reducing computational complexity for weight sequences of quadratic RS functions.

Findings

Proves the ECC for certain rules matrices.

Simplifies computation of weights for quadratic RS functions.

Enables rapid calculation of generating functions.

Abstract

A Boolean function in $n$ variables is rotation symmetric (RS) if it is invariant under powers of $\rho(x_1, \ldots, x_n) = (x_2, \ldots, x_n, x_1)$. An RS function is called monomial rotation symmetric (MRS) if it is generated by applying powers of $\rho$ to a single monomial. The author showed in $2017$ that for any RS function $f_n$ in $n$ variables, the sequence of Hamming weights $wt(f_n)$ for all values of $n$ satisfies a linear recurrence with associated recursion polynomial given by the minimal polynomial of a {\em rules matrix}. Examples showed that the usual formula for the weights $wt(f_n)$ in terms of powers of the roots of the minimal polynomial always has simple coefficients. The conjecture that this is always true is the Easy Coefficients Conjecture (ECC). The present paper proves the ECC if the rules matrix satisfies a certain condition. Major applications include an enormous decrease in the amount of computation that is needed to determine the values of $wt(f_n)$ for a quadratic RS function $f_n$ if either $n$ or the order of the recursion for the weights is large, and a simpler way to determine the Dickson form of $f_n.$ The ECC also enables rapid computation of generating functions which give the values of $wt(f_n)$ as coefficients in a power series.