Multipermutation codes in the Ulam metric

We present a multiset rank modulation scheme capable of correcting translocation errors, motivated by the fact that compared to permutation codes, multipermutation codes offer higher rates and longer block lengths. We show that the appropriate distance measure for code construction is the Ulam metric applied to equivalence classes of permutations, where each permutation class corresponds to a multipermutation. The paper includes a study of multipermutation codes in the Hamming metric, also known as constant composition codes, due to their use in constructing multipermutation codes in the Ulam metric. We derive bounds on the size of multipermutation codes in both the Ulam metric and the Hamming metric, compute their capacity, and present constructions for codes in the Ulam metric based on permutation interleaving, semi-Latin squares, and resolvable Steiner systems.