Recursions for quadratic rotation symmetric functions weights
Thomas W. Cusick
TL;DR
This work addresses how the Hamming weights of quadratic rotation symmetric Boolean functions, and more generally RS/MRS families, satisfy linear recurrences whose structure is governed by a rules matrix. It formalizes the Easy Coefficients Conjecture (ECC), showing that for quadratic RS functions the weight $wt(f_n)$ can be expressed in terms of a sum of powers of roots with very simple coefficients, and proves a special-case version (EECT) under no-root-multiplicity assumptions. The paper provides an explicit quadratic-specific construction of the rules matrix, proves its minimal/characteristic polynomials, and uses this to derive weight recurrences, generating functions, and implications for Dickson form determination. Practically, these results drastically reduce computation for large $n$, enable efficient generation of $wt(f_n)$ via root-sum formulas, and yield rational generating functions for these weights. The approach combines linear-algebraic analysis of the rules matrix with number-theoretic and generating-function techniques to advance understanding of weight recursions in quadratic RS functions.
Abstract
A Boolean function in $n$ variables is rotation symmetric (RS) if it is invariant under powers of $ρ(x_1, \ldots, x_n) = (x_2, \ldots, x_n, x_1)$. An RS function is called monomial rotation symmetric (MRS) if it is generated by applying powers of $ρ$ 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.