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 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.