A class of families of Markov chains defined on the vertices of the n-dimensional hypercube, Ωn={0,1}n, is studied. The single-step transition probabilities Pn,ij, with i,j∈Ωn, are given by \(P_{n,ij}=\frac{(1-{\alpha})^{d_{ij}}}{(2-{\alpha})^{n}}\), where α∈(0,1) and dij is the Hamming distance between i and j. This corresponds to flip independently each component of the vertex with probability \(\frac{1-{\alpha}}{2-{\alpha}}\). The m-step transition matrix \(P_{n,ij}^{m}\) is explicitly computed in a close form. The class is proved to exhibit cutoff. A model-independent result about the vanishing of the first m terms of the expansion in α of \(P_{n,ij}^{m}\) is also proved.