Skew linear recurring sequences of maximal period over Galois rings

M. A. Goltvanitsa ; S. N. Zaitsev ; A. A. Nechaev

Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 3, pp. 5-23

Résumé

Let $p$ be a prime number, $R=GR(q^d,p^d)$ be a Galois ring of $q^d=p^{rd}$ elements and of characteristic $p^d$. Denote by $S=GR(q^{nd},p^d)$ a Galois extension of the ring $R$ of dimension $n$ and by $\check S$ the ring of all linear transformations of the module $_RS$. We call a sequence $v$ over the ring $S$ with the law of recursion $$ \text{for all}\ i\in\mathbb N_0\colon v(i+m)=\psi_{m-1}\bigl(v(i+m-1)\bigr)+\dots+\psi_0\bigl(v(i)\bigr),\quad\psi_0,\dots,\psi_{m-1}\in\check S $$ (i.e., a linear recurring sequence of order $m$ over the module ${}_{\check S}S$) a skew LRS over $S$. It is known that the period $T(v)$ of such a sequence satisfies the inequality $T(v)\le\tau=(q^{nm}-1)p^{d-1}$. If $T(v)=\tau$, then we call $v$ a skew LRS of maximal period (a skew MP LRS) over $S$. A new general characterization of skew MP LRS in terms of coordinate sequences corresponding to some basis of a free module $_RS$ is given. A simple constructive method of building a big enough class of skew MP LRS is stated, and it is proved that the linear complexity of some of them (the rank of the linear recurring sequence) over the module $_SS$ is equal to $mn$, i.e., to the linear complexity over the module $_RS$.