Limited-Magnitude Error-Correcting Gray Codes for Rank Modulation

Abstract

We construct error-correcting codes over permutations under the infinity-metric, which are also Gray codes in the context of rank modulation, i.e., are generated as simple circuits in the rotator graph. These errors model limited-magnitude or spike errors, for which only single-error-detecting Gray codes are currently known. Surprisingly, the error-correcting codes we construct achieve a better asymptotic rate than that of presently known constructions not having the Gray property, and exceed the Gilbert-Varshamov bound. Additionally, we present efficient ranking and unranking procedures, as well as a decoding procedure that runs in linear time. Finally, we also apply our methods to solve an outstanding issue with error-detecting rank-modulation Gray codes (also known in this context as snake-in-the-box codes) under a different metric, the Kendall τ-metric, in the group of permutations over an even number of elements S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2n</sub> , where we provide asymptotically optimal codes.