A criterion for the maximal period property of skew LRS over Galois rings

M. A. Goltvanitsa

MIREA — Russian Technological University, Moscow

Abstract: Let $p$ be a prime number, $R=\mathrm{GR}(q^d,p^d)$ be a Galois ring of cardinality $q^d$ and characteristic $p^d$, where $q = p^r$, $S=\mathrm{GR}(q^{nd},p^d)$ be its extension of degree $n$ and $\check{S}$ be an endomorphism ring of $S$ over $R$. A sequence $v$ with recursion law
$$ \forall i\in\mathbb{N}_0 \colon v(i+m)=\psi_{m-1}(v(i+m-1))+\ldots+\psi_0(v(i)), \psi_0,\ldots,\psi_{m-1 }\in\check{S}, $$
is called skew linear recurrent sequence over $S$ with characteristic polynomial $\Psi(x) = x^m - \sum_{j=0}^{m-1}\psi_jx^j$. The maximal possible period of such sequence is equal to $\tau = (q^{mn} - 1)p^{d-1}$.
In this article we prove a criterion for skew linear recurrent sequence $v$ to achieve maximal possible period in terms of characteristic polynomial of $v$. This criterion generalizes previous known results for so called $\sigma$-splittable skew LRS.

Key words: Galois ring, Frobenius automorphism, ML-sequence, skew LRS, recursion law.

UDC: 519.113.6+512.714+519.719.2

Received 21.V.2025

DOI: 10.4213/mvk506