Cycle lengths in the percolated hypercube

Let Q^d_p be the random subgraph of the d-dimensional binary hypercube obtained after edge-percolation with probability p. It was shown recently by the authors that, for every ε > 0, there is some c = c(ε) > 0 such that, if pd ≥ c, then typically Q^d_p contains a cycle of length at least (1 − ε)2^d . We strengthen this result to show that, under the same assumptions, typically Q^d_p contains cycles of all even lengths between 4 and (1 − ε)2^d.