Primitive Polynomials, Singer Cycles, and Word-Oriented Linear Feedback Shift Registers

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.

💡 Research Summary

The paper investigates the deep interplay between primitive polynomials, Singer cycles in general linear groups, and word‑oriented linear feedback shift registers (σ‑LFSRs). It begins by recalling that a classical LFSR over a finite field 𝔽_q achieves maximal period precisely when its characteristic polynomial is primitive. A σ‑LFSR generalizes this concept by processing the state in word blocks of length m, leading to a transition matrix that is 𝔽_q‑linear but acts on the extended space 𝔽_{q^{m}}. The authors embed this transition matrix into GL_{mn}(𝔽_q) and observe that when the matrix belongs to a Singer cycle—a cyclic subgroup of order q^{mn}−1 that acts transitively on non‑zero vectors—it automatically generates a primitive σ‑LFSR.

Zeng, Han, and He (2007) conjectured a formula for the number of primitive σ‑LFSRs of order n over 𝔽_q, extending the well‑known count of primitive LFSRs (which equals the number of primitive polynomials of degree n). Their conjecture predicts that the number of primitive σ‑LFSRs equals

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