Zero-run lower bound for a linear-feedback shift register

If an output prefix contains $d$ consecutive zeros followed by a one, no register of degree at most $d$ can generate it: the all-zero state would have been reached and would remain zero forever.

The seven consecutive zeros followed by one force degree at least eight. Degree eight is attained by $s_{n+8}=s_n$, with initial state $10000000$ and feedback polynomial $x^8+1$.