Primitive Polynomials, Singer Cycles, and Word-Oriented Linear Feedback Shift Registers

TL;DR

This paper investigates the enumeration of primitive word-oriented linear feedback shift registers using Singer cycles, proving a special case of a conjecture and proposing a method for the general case, linking finite field structures and open problems.

Contribution

It proves a special case of a conjecture on primitive σ-LFSRs using Singer cycles and outlines a potential approach for the general case, connecting to open questions in finite field theory.

Findings

Confirmed the conjecture in a specific case using Singer cycles.

Outlined a plausible approach for the general conjecture.

Connected the problem to open questions on splitting subspaces.

Abstract

Using the structure of Singer cycles in general linear groups, we prove that a conjecture of Zeng, Han and He (2007) holds in the affirmative in a special case, and outline a plausible approach to prove it in the general case. This conjecture is about the number of primitive $\sigma$-LFSRs of a given order over a finite field, and it generalizes a known formula for the number of primitive LFSRs, which, in turn, is the number of primitive polynomials of a given degree over a finite field. Moreover, this conjecture is intimately related to an open question of Niederreiter (1995) on the enumeration of splitting subspaces of a given dimension.