Primitive polynomials, singer cycles and word-oriented linear feedback shift registers

Abstract

Abstract. 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 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 (1995) on the enumeration of splitting subspaces of a given dimension.

On the number of irreducible linear transformation shift registers

“…The exact number of primitive LFSRs over a finite field of a fixed order is known. For σ -LFSRs, the following conjecture was proposed in the binary case in [18] and extended to the q-ary case in [5].…”

Скошенные ЛРП Максимального Периода Над Кольцами Галуа

“…This conjecture was checked for many values m, n, q, and proved in special cases: for m = 1 and any n, q (in [11]), for n = 2 and any m, q (in [12]). …”

An asymptotic formula for the number of irreducible transformation shift registers

“…We refer to [10] and [11] for recent progress on this conjecture and to [4] for a proof of this conjecture. It is also known from [11] and [4], see also [18], that the number of irreducible σ-LFSRs is (4) 1 mn q m(m−1)(n−1)…”