A permutation code of length n and distance d is a set of permutations from some fixed set of n symbols such that the Hamming distance between each distinct x, y ∈ is at least d. In this note, we determine some new results on the maximum size of a permutation code with distance equal to 4, the smallest interesting value. The upper bound is improved for almost all n via an optimization problem on Young diagrams. A new recursive construction improves known lower bounds for small values of n.