An asymptotic formula for the number of irreducible transformation shift registers

TL;DR

This paper derives an asymptotic formula for counting irreducible transformation shift registers and provides a concise proof for the exact count at order two, advancing understanding in sequence generator enumeration.

Contribution

It introduces an asymptotic formula for irreducible transformation shift registers and simplifies the proof for the order two case using recent theoretical generalizations.

Findings

Asymptotic formula for special cases of irreducible transformation shift registers

Short proof for the exact number at order two

Enhanced understanding of sequence generator enumeration

Abstract

We consider the problem of enumerating the number of irreducible transformation shift registers. We give an asymptotic formula for the number of irreducible transformation shift registers in some special cases. Moreover, we derive a short proof for the exact number of irreducible transformation shift registers of order two using a recent generalization of a theorem of Carlitz.