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Enumeration of linear transformation shift registers

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Abstract

We consider the problem of counting the number of linear transformation shift registers (TSRs) of a given order over a finite field. We derive explicit formulae for the number of irreducible TSRs of order two. An interesting connection between TSRs and self-reciprocal polynomials is outlined. We use this connection and our results on TSRs to deduce a theorem of Carlitz on the number of self-reciprocal irreducible monic polynomials of a given degree over a finite field.

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Notes

  1. By ‘order’ of a TSR, we mean the order of the recurrence relation defining the TSR, not the multiplicative order of the corresponding state transition matrix (if and when it lies in \(\mathrm{GL }_{mn}({\mathbb F}_q)\)). We follow this convention throughout.

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Acknowledgments

The author would like to thank Prof. Sudhir Ghorpade for suggesting the problem, and Sartaj Ul Hasan and Prof. Gilles Lachaud for their comments on a preliminary draft of this paper. This research was partly carried out at the Indian Statistical Institute, Bangalore and partly at the Institut de Mathématiques de Luminy, Marseille.

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  1. Institut de Mathématiques de Luminy, Luminy Case 907, 13288, Marseille Cedex 9, France

    Samrith Ram

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Communicated by D. Panario.

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Ram, S. Enumeration of linear transformation shift registers. Des. Codes Cryptogr. 75, 301–314 (2015). https://doi.org/10.1007/s10623-013-9913-5

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  • DOI: https://doi.org/10.1007/s10623-013-9913-5

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