Systematic Error-Correcting Codes for Rank Modulation

The rank-modulation scheme has been recently proposed for efficiently storing data in nonvolatile memories. In this paper, we explore [n,k,d] systematic error-correcting codes for rank modulation. Such codes have length n, k information symbols, and minimum distance d. Systematic codes have the benefits of enabling efficient information retrieval in conjunction with memory-scrubbing schemes. We study systematic codes for rank modulation under Kendall's $\tau $ -metric as well as under the $\ell -\infty $ -metric. In Kendall's $\tau $ -metric, we present [k+2,k,3] systematic codes for correcting a single error, which have optimal rates, unless systematic perfect codes exist. We also study the design of multierror-correcting codes, and provide a construction of [k+t+1,k,2t+1] systematic codes, for large-enough k. We use nonconstructive arguments to show that for rank modulation, systematic codes achieve the same capacity as general error-correcting codes. Finally, in the $\ell -\infty $ -metric, we construct two [n,k,d] systematic multierror-correcting codes, the first for the case of d=O(1) and the second for d=\Theta (n). In the latter case, the codes have the same asymptotic rate as the best codes currently known in this metric.