Fast mixing of a randomized shift-register Markov chain

We present a Markov chain on the $n$-dimensional hypercube ${0,1}^n$ which satisfies $t_{{rm mix}}(epsilon) = n[1 + o(1)]$. This Markov chain alternates between random and deterministic moves and we prove that the chain has cut-off with a window of size at most $O(n^{0.5+delta})$ where $delta>0$. The deterministic moves correspond to a linear shift register.