Fast mixing of a randomized shift-register Markov chain

We present a Markov chain on the n-dimensional hypercube {0, 1} which satisfies t (n) mix(ε) = n[1 + o(1)]. This Markov chain alternates between random and deterministic moves and we prove that the chain has cutoff with a window of size at most O(n0.5+δ) where δ > 0. The deterministic moves correspond to a linear shift register.