Rank permutation group codes based on Kendall's correlation statistic

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-errordetecting 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 τ-metric, in the group of permutations over an even number of elements S 2n , where we provide asymptotically optimal codes.

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“…4,7] For all n 1 there exist G ↑ (n, n!) codes, that is, complete and cyclic "push-to-the-top" Gray codes over the symmetric group S n .…”

Section: Lemma 4 Ifmentioning

confidence: 99%

“…Since j = 1, we defineσ ↾Ŵ The codeword σ = [11,1,8,6,7,2,12,13,3,5,9,14,4,10,15] appears in the code generated in this case, as can be seen by identifying its C 5 , C 4 , C 3 , and C 2 parents as, respectively, 6,11,1,7,12,2,8,13,3,9,14,4,5,10,15] , σ 4 = [6,11,1,7,12,2,8,13,3,5,9,14,4,10,15] , 6,11,1,8,7,2,12,...…”

Section: Theorem 29mentioning

confidence: 99%

Limited-magnitude error-correcting Gray codes for rank modulation

Yehezkeally

Schwartz

2016

2016 IEEE International Symposium on Information Theory (ISIT)

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“…Rank modulation has been studied intermittently since the early works of Slepian [17] (later extended in [1]), in which permutations were used to digitize vectors from a time-discrete memoryless Gaussian source, and Chadwick and Kurz [5], in which permutations were used in the context of signal detection over channels with non-Gaussian noise (especially impulse noise). Other works on the subject include [1]- [4], [6], [7].…”

Section: Introductionmentioning

confidence: 99%

Generalized Gray Codes for Local Rank Modulation

Gad

Langberg

Schwartz

et al. 2013

IEEE Trans. Inform. Theory

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Abstract-We consider the local rank-modulation scheme in which a sliding window going over a sequence of real-valued variables induces a sequence of permutations. Local rank-modulation is a generalization of the rankmodulation scheme, which has been recently suggested as a way of storing information in flash memory.We study Gray codes for the local rank-modulation scheme in order to simulate conventional multi-level flash cells while retaining the benefits of rank modulation. Unlike the limited scope of previous works, we consider code constructions for the entire range of parameters including the code length, sliding window size, and overlap between adjacent windows. We show our constructed codes have asymptotically-optimal rate. We also provide efficient encoding, decoding, and next-state algorithms.

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“…Codes over permutations are also referred to as permutation arrays and have been studied in the past under different metrics [2], [3], [6], [7], [9], [10], [16]. Specifically, permutation arrays under the ℓ ∞ -metric were considered in [14].…”

mentioning

confidence: 99%

On the Labeling Problem of Permutation Group Codes Under the Infinity Metric

Tamo

Schwartz

2012

IEEE Trans. Inform. Theory

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Abstract-Codes over permutations under the infinity norm have been recently suggested as a coding scheme for correcting limited-magnitude errors in the rank modulation scheme. Given such a code, we show that a simple relabeling operation, which produces an isomorphic code, may drastically change the minimal distance of the code. Thus, we may choose a code structure for efficient encoding/decoding procedures, and then optimize the code's minimal distance via relabeling.We formally define the relabeling problem, and show that all codes may be relabeled to get a minimal distance at most 2. The decision problem of whether a code may be relabeled to distance 1 is shown to be NP-complete, and calculating the best achievable minimal distance after relabeling is proved hard to approximate.Finally, we consider general bounds on the relabeling problem. We specifically show the optimal relabeling distance of cyclic groups. A specific case of a general probabilistic argument is used to show AGL(p) may be relabeled to a minimal distance of p − O( p ln p).

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Rank permutation group codes based on Kendall's correlation statistic

Cited by 46 publications

References 4 publications

Limited-magnitude error-correcting Gray codes for rank modulation

Limited-magnitude error-correcting Gray codes for rank modulation

Generalized Gray Codes for Local Rank Modulation

On the Labeling Problem of Permutation Group Codes Under the Infinity Metric

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