An Enumeration Algorithm for Binary Coprime Polynomials with Nonzero Constant Term

TL;DR

This paper presents a combinatorial algorithm for enumerating coprime polynomial pairs over  with nonzero constant terms, leveraging algebraic language theory and Euclid's algorithm to count and generate such pairs.

Contribution

It introduces a novel enumeration algorithm based on the analysis of quotient sequences, combining language theory and combinatorics for coprime polynomial pairs.

Findings

Sequences of constant terms form a regular language.

Sequences of degrees correspond to compositions of natural numbers.

Intermediate terms can be chosen freely, enabling enumeration.

Abstract

We address the enumeration of coprime polynomial pairs over $\F_2$ where both polynomials have a nonzero constant term, motivated by the construction of orthogonal Latin squares via cellular automata. To this end, we leverage on Benjamin and Bennett's bijection between coprime and non-coprime pairs, which is based on the sequences of quotients visited by dilcuE's algorithm (i.e. Euclid's algorithm ran backward). This allows us to break our analysis of the quotients in three parts, namely the enumeration and count of: (1) sequences of constant terms, (2) sequences of degrees, and (3) sequences of intermediate terms. For (1), we show that the sequences of constant terms form a regular language, and use classic results from algebraic language theory to count them. Concerning (2), we remark that the sequences of degrees correspond to compositions of natural numbers, which have a simple combinatorial description. Finally, we show that for (3) the intermediate terms can be freely chosen. Putting these three obeservations together, we devise a combinatorial algorithm to enumerate all such coprime pairs of a given degree, and present an alternative derivation of their counting formula.