The R-transform as a power map and its generalisations to higher degree

TL;DR

This paper generalizes Cohen's R-transform to higher degrees, providing iterative methods to construct irreducible polynomials over finite fields with specific degrees, and explores their properties and extensions.

Contribution

It introduces a generalized R-transform for arbitrary degrees, extending previous constructions and unifying various recursive methods for polynomial generation over finite fields.

Findings

Generalized R-transform for degrees t > 2

Iterative constructions for irreducible polynomials of degree nt^r

Extensions to quadratic fields and connections to existing recursive methods

Abstract

We give iterative constructions for irreducible polynomials over F_q of degree nt^r for all nonnegative integers r, starting from irreducible polynomials of degree n. The iterative constructions correspond modulo fractional linear transformations to compositions with power functions x^t. The R-transform introduced by Cohen is recovered as a particular case corresponding to x^2, hence we obtain a generalization of Cohen's R-transform (t=2) to arbitrary degrees t bigger that two. Important properties like self-reciprocity and invariance of roots under certain automorphisms are deduced from invariance under multiplication by appropriate roots of unity. Extending to quadratic extensions of F_q we recover and generalize a recently obtained recursive construction of Panario, Reis and Wang.