An Algorithm for Computing Minimal Bidirectional Linear Recurrence Relations

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 nonzero. Such a recurrence relation can then be used to generate the sequence in both directions (increasing or decreasing order of indices), so we 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 describe the set of all solutions to this problem and show that if a sequence admits more than one linear recurrence relation then it admits a bidirectional one. We also prove some properties regarding the bidirectionality of the recurrences of the prefixes of the sequence.