We consider the problem of enumerating irreducible transformation shift registers. We give an asymptotic formula for the number of irreducible transformation shift registers in some special cases. Moreover, we derive a short proof for the exact number of irreducible transformation shift registers of order two using a recent generalization of a theorem of Carlitz.

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Keywords Block companion matrix, Characteristic polynomial, Galois group, Irreducible polynomial, Primitive polynomial, Transformation shift register
Persistent URL dx.doi.org/10.1016/j.laa.2015.06.020
Journal Linear Algebra and Its Applications
Citation
Cohen, S.D. (Stephen D.), Hasan, S.U. (Sartaj Ul), Panario, D, & Wang, Q. (2015). An asymptotic formula for the number of irreducible transformation shift registers. Linear Algebra and Its Applications, 484, 46–62. doi:10.1016/j.laa.2015.06.020