Systematic Error-Correcting Codes for Rank Modulation

• Published in: IEEE transactions on information theory Vol. 61; no. 1; pp. 17 - 32
• Main Authors: Hongchao Zhou, Schwartz, Moshe, Jiang, Anxiao Andrew, Bruck, Jehoshua
• Format: Journal Article
• Language: English
• Published: New York IEEE 01-01-2015; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: <italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"> <inline-formula> <tex-math notation="LaTeX"> ell _\infty </tex-math> </inline-formula> -metric; Codes; Error correction & detection; Error correction codes; error-correcting codes; Flash memory; Information retrieval; Information theory; Kendall’s <italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"> <inline-formula> <tex-math notation="LaTeX"> tau </tex-math> </inline-formula> -metric; Measurement; metric embeddings; Modulation; permutations; rank modulation; Redundancy; systematic codes; Systematics; Tin; Vectors; Kendall’s tau -metric; error-correcting codes; ell _\infty -metric; Flash memory; rank modulation; permutations; systematic codes; metric embeddings
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2014.2365499

The rank-modulation scheme has been recently proposed for efficiently storing data in nonvolatile memories. In this paper, we explore [n, k, d] systematic error-correcting codes for rank modulation. Such codes have length n, k information symbols, and minimum distance d. Systematic codes have the benefits of enabling efficient information retrieval in conjunction with memory-scrubbing schemes. We study systematic codes for rank modulation under Kendall's T-metric as well as under the ℓ ∞ -metric. In Kendall's T-metric, we present [k + 2, k, 3] systematic codes for correcting a single error, which have optimal rates, unless systematic perfect codes exist. We also study the design of multierror-correcting codes, and provide a construction of [k + t + 1, k, 2t + 1] systematic codes, for large-enough k. We use nonconstructive arguments to show that for rank modulation, systematic codes achieve the same capacity as general error-correcting codes. Finally, in the ℓ ∞ -metric, we construct two [n, k, d] systematic multierror-correcting codes, the first for the case of d = 0(1) and the second for d = Θ(n). In the latter case, the codes have the same asymptotic rate as the best codes currently known in this metric.