An iterative algorithm for parametrization of shortest length linear shift registers over finite chain rings

Abstract

The construction of shortest feedback shift registers for a finite sequence S1, . . . , SN is considered over the finite ring Zpr . A novel algorithm is presented that yields a parametrization of all shortest feedback shift registers for the sequence of numbers S1, . . . , SN , thus solving an open problem in the literature. The algorithm iteratively processes each number, starting with S1, and constructs at each step a particular type of minimal Gröbner basis. The construction involves a simple update rule at each step which leads to computational efficiency. It is shown that the algorithm simultaneously computes a similar parametrization for the reciprocal sequence SN , . . . , S1. * M. Kuijper is with the

Noncatastrophic convolutional codes over a finite ring

“…We note that different definitions have been considered in the literature, see for instance [4,10,11,21]. The nonfree case lies beyond the scope of this work but it can also be treated using the theory of p-basis and p-generating sequences, see for instance [11,12,18,21].…”

State representations of convolutional codes over a finite ring

“…Most of the large body of literature on convolutional codes and on the relation of these codes with linear systems has been devoted to the field case. But sometimes it is too restrictive to consider fields and so, part of this theory has been extended to finite rings [8,16,18,19,20,25,33,41]. This work continues this thread of research and we aim at studying convolutional codes over the ring Z p r (where p is a prime and r is an integer) from a system theoretical point of view.…”