Systematic Error-Correcting Codes for Permutations and Multi-Permutations

Abstract

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.