Introduction to finite fields and their applications
Introduction to finite fields and their applications
On the 2-Adic Complexity and the k-Error 2 -Adic Complexity of Periodic Binary Sequences
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Minimal polynomial algorithms for finite sequences
IEEE Transactions on Information Theory
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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.