Mixing rates for a random walk on the cube

For a simple random walk on the cube a coupling and a strong uniform time are given. The coupling gives an upper bound on the variation distance between the distribution after k steps and the uniform distribution that is almost the best possible. The strong uniform time is used to calculate the variation distance and the separation. The coupling and strong uniform time are intimately related to a hitting time for the Ehrenfest chain and the time taken by a random graph to become connected, respectively.