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The rank-modulation scheme has been recently proposed for efficiently storingdata in nonvolatile memories. Error-correcting codes are essential for rankmodulation, however, existing results have been limited. In this work weexplore a new approach, \emph{systematic error-correcting codes for rankmodulation}. Systematic codes have the benefits of enabling efficientinformation retrieval and potentially supporting more efficient encoding anddecoding procedures. We study systematic codes for rank modulation underKendall's $\tau$-metric as well as under the $\ell_\infty$-metric.  In Kendall's $\tau$-metric we present $[k+2,k,3]$-systematic codes forcorrecting one error, which have optimal rates, unless systematic perfect codesexist. We also study the design of multi-error-correcting codes, and providetwo explicit constructions, one resulting in $[n+1,k+1,2t+2]$ systematic codeswith redundancy at most $2t+1$. We use non-constructive arguments to show theexistence of $[n,k,n-k]$-systematic codes for general parameters. Furthermore,we prove that for rank modulation, systematic codes achieve the same capacityas general error-correcting codes.  Finally, in the $\ell_\infty$-metric we construct two $[n,k,d]$ systematicmulti-error-correcting codes, the first for the case of $d=O(1)$, and thesecond for $d=\Theta(n)$. In the latter case, the codes have the sameasymptotic rate as the best codes currently known in this metric.

Systematic Error-Correcting Codes for Rank Modulation