Enumeration of Linear Transformation Shift Registers

TL;DR

This paper derives explicit formulas for counting irreducible linear transformation shift registers over finite fields, explores their connection to self-reciprocal polynomials, and applies these results to prove a theorem on such polynomials.

Contribution

It provides the first explicit enumeration formulas for irreducible TSRs of order two and links TSRs to self-reciprocal polynomials, extending existing theoretical knowledge.

Findings

Derived explicit formulas for irreducible TSRs of order two

Established a connection between TSRs and self-reciprocal polynomials

Deduced a theorem of Carlitz on self-reciprocal irreducible polynomials

Abstract

We consider the problem of counting the number of linear transformation shift registers (TSRs) of a given order over a finite field. We derive explicit formulae for the number of irreducible TSRs of order two. An interesting connection between TSRs and self-reciprocal polynomials is outlined. We use this connection and our results on TSRs to deduce a theorem of Carlitz on the number of self-reciprocal irreducible monic polynomials of a given degree over a finite field.