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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Communicated by J. Bierbrauer.
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Ghorpade, S.R., Hasan, S.U. & Kumari, M. Primitive polynomials, singer cycles and word-oriented linear feedback shift registers. Des. Codes Cryptogr. 58, 123–134 (2011). https://doi.org/10.1007/s10623-010-9387-7
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DOI: https://doi.org/10.1007/s10623-010-9387-7
Keywords
- Primitive polynomial
- Linear Feedback Shift Register (LFSR)
- Primitive recursive vector sequence
- Singer cycle
- Singer subgroup
- Splitting subspaces