  We introduce a general class of random walks on the $N$-hypercube, study
cut-off for the mixing time, and provide several types of representation for
the transition probabilities. We observe that for a sub-class of these
processes with long range (i.e. non-local) there exists a critical value of the
range that allows an "almost-perfect" mixing in at most three steps. In other
words, the total variation distance between the three steps transition and the
stationary distribution decreases geometrically in $N$, which is the dimension
of the hypercube. In some cases, the walk mixes almost-perfectly in exactly two
steps. Notice that a well-known result (Theorem 1 in Diaconis and Shahshahani
(1986)) shows that there exist no random walk on Abelian groups (such as the
hypercube) which mixes perfectly in exactly two steps.
