An algorithm for computing minimal bidirectional linear recurrence relations

Abstract

We consider the problem of computing a linear recurrence relation (or equivalently a Linear Feedback Shift Register) of minimum order for a finite sequence over a field, with the additional requirement that not only the highest but also the lowest coefficient of the recurrence is non-zero. Such a recurrence relation can then be used to generate the sequence in both directions (increasing or decreasing order of indices), so we will call it bidirectional. If the field is finite, a sequence is periodic if and only if it admits a bidirectional linear recurrence relation. For solving the above problem we propose an algorithm similar to the Berlekamp-Massey algorithm and prove its correctness. We also describe the set of all solutions to this problem and prove some properties of the minimal polynomials of initial segments of the sequence.