Primitive polynomials,singer cycles and word-oriented linear feedback shift registers |
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Authors: | Email author" target="_blank">Sudhir?R?GhorpadeEmail author Sartaj?Ul?Hasan Meena?Kumari |
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Institution: | 1.Department of Mathematics,Indian Institute of Technology Bombay,Mumbai,India;2.Scientific Analysis Group, Defense Research and Development Organisation,Delhi,India |
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Abstract: | Using the structure of Singer cycles in general linear groups, we prove that a conjecture of Zeng et al. (Word-Oriented Feedback
Shift Register: σ-LFSR, 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 σ-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 (Finite Fields Appl 1:3–30, 1995) on the enumeration of splitting subspaces of
a given dimension. |
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