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arXiv · 1601.05218

Limited-Magnitude Error-Correcting Gray Codes for Rank Modulation

Yonatan Yehezkeally·Moshe Schwartz
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Abstract

We construct Gray codes over permutations for the rank-modulation scheme, which are also capable of correcting errors under the infinity-metric. 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 (snake-in-the-box codes) under a different metric, the Kendall $\tau$-metric, in the group of permutations over an even number of elements $S_{2n}$, where we provide asymptotically optimal codes.

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Yonatan Yehezkeally, Moshe Schwartz. 2016-01-20. Limited-Magnitude Error-Correcting Gray Codes for Rank Modulation. https://doi.org/10.1109/tit.2017.2719710

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