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 $S_1,\ldots ,S_N$ is considered over finite chain rings, such as ${\mathbb Z}_{p^r}$. A novel algorithm is presented that yields a parametrization of all shortest feedback shift registers for the sequence of numbers $S_1,\ldots ,S_N$, thus solving an open problem in the literature. The algorithm iteratively processes each number, starting with $S_1$, and constructs at each step a particular type of minimal 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 reverse sequence $S_N,\ldots ,S_1$. The complexity order of the algorithm is shown to be $O(r N^2)$.