1. Primitive Polynomials, Singer Cycles, and Word-Oriented Linear Feedback Shift Registers
- Author
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Ghorpade, Sudhir R., Hasan, Sartaj Ul, and Kumari, Meena
- Subjects
Mathematics - Combinatorics ,Computer Science - Information Theory ,11T06, 11T31, 20G40, 94A60 - 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., Comment: Version 2 with some minor changes; to appear in Designs, Codes and Cryptography.
- Published
- 2009
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