Permutation Codes in Levenshtein, Ulam and Generalized Kendall-Tau Metrics

Our main goal in this work is to study permutation codes in the Levenshtein metric which can correct multiple deletions. Permutation codes in the Levenshtein metric are known to have a close relationship with those in Ulam and generalized Kendall-tau metrics. In this work, we present a new connection between different kinds of distances over two permu-tations' including Hamming, Levenshtein, Ulam, and generalized Kendall-tau distance. From this new connection and some known permutation codes in the Hamming metric, we obtain new permu-tation codes in Levenshtein, Ulam, and generalized Kendall-tau metrics with better size. In particular, we show that there exist permutation codes of length $n$ for correcting $t$ deletions with at most $(3t+1)\log n+o(\log n)$ bits of redundancy. Furthermore, we present the construction of our permutation codes for correcting $t$ deletions with a specific decoding process.