Skew linear recurring sequences of maximal period over galois rings

• DOI: 10.1007/s10958-012-1054-2
• Publication Date: 2012-10-18T21:31:59Z
• Authors: M. A. Goltvanitsa; S. N. Zaitsev; A. A. Nechaev
• Keywords: 0101 mathematics; 01 natural sciences

Let p be a prime number, and R = GR(qd, pd) be a Galois ring of qd = prd elements and of characteristic pd. Denote by S = GR(qnd, pd) a Galois extension of the ring R of dimension n and by 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 $$ {\mathrm{for}\ \mathrm{all}\ }i \in {\mathbb{N}_0}:v\left( {i + m} \right) = {\psi_{m - 1}}\left( {v\left( {i + m - 1} \right)} \right) + \cdots + {\psi_0}\left( {v(i)} \right),\quad {\psi_0}, \ldots, {\psi_{m - 1}} \in \textit{\v{S}} $$ (i.e., a linear recurring sequence of order m over the module SS) a skew LRS over S. It is known that the period T(v) of such a sequence satisfies the inequality T(v) ≤ τ = (qnm−1)pd−1. If T(v) = τ , 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.