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A binary linear-feedback shift register of length $L$ stores $L$ bits. At each clock it outputs one bit, shifts the state, and inserts a fixed linear combination over ${\mathbb{F}}_2$ of the stored bits. Equivalently, its output obeys a recurrence

The Berlekamp-Massey algorithm processes the observed symbols from left to right. It maintains the shortest current connection polynomial $C(D)$, its degree $L$, and a saved polynomial from the most recent degree increase. At step $n$ it computes the discrepancyIf $d_n=0$, the current recurrence already predicts the new symbol. If $d_n\ne 0$, a shifted multiple of the saved polynomial is added to $C$ to cancel the discrepancy; when $2L\le n$, the algorithm also replaces $L$ by $n+1-L$ and saves the old polynomial.

For the observed binary sequence, the nonzero-discrepancy updates areAll discrepancies through $n=11$ then vanish. Thus the shortest recurrence isor, equivalently, $s_{n+4}=s_{n+1}+s_n$. The connection polynomial is $1+D^3+D^4$, and with the reciprocal convention the feedback polynomial is