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This article is cited in 7 scientific papers (total in 7 papers)
Reconstruction of a linear recurrence of maximal period over a Galois ring from its highest coordinate sequence
A. S. Kuzmin, A. A. Nechaev
Abstract:
Let $R=GR(q^n,q^n)$ be a Galois ring of cardinality $q^n$ and characteristic $p^n$, $q=p^r$, $p$ be a prime. We call a subset $K\subset R$ a coordinate set if $0\in K$ and for any $a\in R$ there exists a unique $\varkappa(a)\in K$ such that $a\equiv\varkappa(a)\pmod{pR}$. Let $u$ be a linear recurring sequence of maximal period (MP LRS) over a ring $R$. Then any its term $u(i)$ admits a unique representation in the form
$$
u(i)=w_0(i)+pw_1(i)+\dots+p^{n-1}w_{n-1}(i),\qquad w_t(i)\in K,\quad t\in\{0,\dots,n-1\}.
$$
We pose the following conjecture: the sequence $u$ can be uniquely reconstructed from the sequence $w_{n-1}$ for any choice of the coordinate set $K$. It is proved that such a reconstruction is possible under some conditions on $K$. In particular, it is possible for any $K$ if $R=\mathbf Z_{p^n}$ and for any Galois ring $R$ if $K$ is a $p$-adic (Teichmüller) coordinate set.
Received: 16.04.2010
Citation:
A. S. Kuzmin, A. A. Nechaev, “Reconstruction of a linear recurrence of maximal period over a Galois ring from its highest coordinate sequence”, Diskr. Mat., 23:2 (2011), 3–31; Discrete Math. Appl., 21:2 (2011), 145–178
Linking options:
https://www.mathnet.ru/eng/dm1137https://doi.org/10.4213/dm1137 https://www.mathnet.ru/eng/dm/v23/i2/p3
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