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

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)).