Limited-Magnitude Error-Correcting Gray Codes for Rank Modulation

TL;DR

This paper introduces new error-correcting Gray codes for rank modulation that outperform existing codes in rate, include efficient encoding/decoding algorithms, and address open problems in permutation error detection under multiple metrics.

Contribution

It constructs novel error-correcting Gray codes for permutations with improved asymptotic rates and efficient algorithms, surpassing known bounds and solving open problems.

Findings

Codes exceed Gilbert-Varshamov bound

Decoding runs in linear time

Asymptotically optimal codes for Kendall tau-metric

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.