A Note on the Games-Chan Algorithm

TL;DR

This paper generalizes the Games-Chan algorithm from binary to q-ary sequences, enabling efficient computation of minimal periods and polynomial root multiplicities using generating functions and polynomials.

Contribution

It extends the Games-Chan algorithm to q-ary sequences and introduces a method to find root multiplicities in polynomials efficiently.

Findings

Generalized the algorithm to q-ary sequences

Developed a polynomial-based method for root multiplicity

Achieved logarithmic iteration complexity

Abstract

The Games-Chan algorithm finds the minimal period of a periodic binary sequence of period $2^n$, in $n$ iterations. We generalise this to periodic $q$-ary sequences (where $q$ is a prime power) using generating functions and polynomials and apply this to find the multiplicity of $x-1$ in a $q$-ary polynomial $f$ in $\log_{\,q}\deg(f)$ iterations.