Primitive Polynomials, Singer Cycles, and Word-Oriented Linear Feedback Shift Registers
Combinatorics
2011-02-10 v2 Information Theory
math.IT
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 -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.
Keywords
Cite
@article{arxiv.0904.1331,
title = {Primitive Polynomials, Singer Cycles, and Word-Oriented Linear Feedback Shift Registers},
author = {Sudhir R. Ghorpade and Sartaj Ul Hasan and Meena Kumari},
journal= {arXiv preprint arXiv:0904.1331},
year = {2011}
}
Comments
Version 2 with some minor changes; to appear in Designs, Codes and Cryptography.