James L. Massey showed that Berlekamp's iterative BCH-decoding algorithm solves a general register-synthesis problem: find a shortest linear-feedback shift register generating a prescribed finite sequence. His 1969 paper derives the algorithm through the lengths and discrepancies of successive candidate recurrences, connecting the Berlekamp-Massey algorithm to error correction.[ref]
A sequence that satisfies can be generated from stored symbols and feedback coefficients over a field. A newly observed symbol may contradict the current recurrence; the algorithm then updates the candidate using a prior discrepancy. Minimal synthesis is therefore more than finding some recurrence that happens to fit the observations.
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