Systematic Error-Correcting Codes for Permutations and Multi-Permutations

Multi-permutations and in particular permutations appear in various applications in an information theory. New applications, such as rank modulation for flash memories, have suggested the need to consider error-correcting codes for multi-permutations. In this paper, we study systematic error-correcting codes for multi-permutations in general and for permutations in particular. For a given number of information symbols k, and for any integer t, we present a construction of (k+r,k)systematic t-error-correcting codes, for permutations of length k+r, where the number of redundancy symbols r is relatively small. In particular, for a given t and for sufficiently large k, we obtain r=t+1, while a lower bound on the number of redundancy symbols is shown to be t. The same construction is also applied to obtain related systematic error-correcting codes for any types of multi-permutations.