Single-deletion-correcting codes over permutations

Motivated by the rank modulation scheme for flash memories, we consider an information representation system with relative values (permutations) and study codes for correcting deletions. In contrast to the case of a deletion in a regular (with absolute values) representation system, a deletion in this new paradigm results in a new permutation over the remaining symbols. For example, the deletion of 3 (or 2) from (1, 3, 2, 4) yields (1, 2, 3); while the deletion of 1 yields (2, 1, 3). Codes for correcting deletions in permutations were studied by Levenshtein under a different model, however, he considered absolute values where the deletions are missing symbols. We study the single deletion relative-values model and prove that a code can correct a single deletion if and only if it can correct a single insertion. Using the concept of a signature of a permutation, we construct single-deletion correcting codes and prove that they are asymptotically optimal with respect to an upper bound that we derive. Finally, we describe an efficient decoding algorithm.