Limited-magnitude error-correcting gray codes for rank modulation

Yonatan Yehezkeally, Moshe Schwartz

Research output: Contribution to journalArticlepeer-review

8 Scopus citations

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 $S-{2n}$ , where we provide asymptotically optimal codes.

Original languageEnglish
Article number7959111
Pages (from-to)5774-5792
Number of pages19
JournalIEEE Transactions on Information Theory
Volume63
Issue number9
DOIs
StatePublished - 1 Sep 2017

Keywords

  • Gray codes
  • error-correcting codes
  • permutations
  • rank modulation
  • spread-d circuit codes

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