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

TL;DR

This paper introduces an efficient iterative algorithm over finite rings to parametrize all shortest feedback shift registers for a sequence, solving an open problem and improving computational methods in the field.

Contribution

The paper presents a novel iterative algorithm that constructs all shortest feedback shift registers over finite rings, including a minimal Gr"obner basis update rule, addressing an open problem.

Findings

Algorithm computes all shortest shift registers for sequences over Z_{p^r}

Constructs minimal Gr"obner bases efficiently at each step

Simultaneously parametrizes reciprocal sequences

Abstract

The construction of shortest feedback shift registers for a finite sequence S_1,...,S_N is considered over the finite ring 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,...,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 Gr\"obner 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 S_N,...,S_1.