We investigate error-correcting codes for a the rank-modulation scheme with an application to flash memory devices. In this scheme, a set of n cells stores information in the permutation induced by the different charge levels of the individual cells. The resulting scheme eliminates the need for discrete cell levels, overcomes overshoot errors when programming cells (a serious problem that reduces the writing speed), and mitigates the problem of asymmetric errors. In this paper, we study the properties of error-correcting codes for charge-constrained errors in the rank-modulation scheme. In this error model the number of errors corresponds to the minimal number of adjacent transpositions required to change a given stored permutation to another erroneous one-a distance measure known as Kendall's ? -distance. We show bounds on the size of such codes, and use metric-embedding techniques to give constructions which translate a wealth of knowledge of codes in the Lee metric to codes over permutations in Kendall's ?-metric. Specifically, the one-error-correcting codes we construct are at least half the ball-packing upper bound.

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Correcting Charge-Constrained Errors in the Rank-Modulation Scheme

We study error-correcting codes for permutations under the infinity norm, motivated by a novel storage scheme for flash memories called rank modulation. In this scheme, a set of n flash cells are combined to create a single virtual multilevel cell. Information is stored in the permutation induced by the cell charge levels. Spike errors, which are characterized by a limited-magnitude change in cell charge levels, correspond to a low-distance change under the infinity norm. We define codes protecting against spike errors, called limited-magnitude rank-modulation codes (LMRM codes), and present several constructions for these codes, some resulting in optimal codes. These codes admit simple recursive, and sometimes direct, encoding and decoding procedures. We also provide lower and upper bounds on the maximal size of LMRM codes both in the general case, and in the case where the codes form a subgroup of the symmetric group. In the asymptotic analysis, the codes we construct outperform the Gilbert-Varshamov-like bound estimate.

Correcting Limited-Magnitude Errors in the Rank-Modulation Scheme

Rank modulation is a way of encoding information to correct errors in flash memory devices as well as impulse noise in transmission lines. Modeling rank modulation involves construction of packings of the space of permutations equipped with the Kendall tau distance. As our main set of results, we present several general constructions of codes in permutations that cover a broad range of code parameters. In particular, we show a number of ways in which conventional error-correcting codes can be modified to correct errors in the Kendall space. Our constructions are nonasymptotic and afford simple encoding and decoding algorithms of essentially the same complexity as required to correct errors in the Hamming metric. As an example, from binary Bose-Chaudhuri-Hocquenghem codes, we obtain codes correcting t Kendall errors in n memory cells that support the order of n!/(log2n!)t messages, for any constant t=1,2,.... We give many examples of rank modulation codes with specific parameters. Turning to asymptotic analysis, we construct families of rank modulation codes that correct a number of errors that grows with n at varying rates, from Θ(n) to Θ(n2). One of our constructions gives rise to a family of rank modulation codes for which the tradeoff between the number of messages and the number of correctable Kendall errors approaches the optimal scaling rate.

Constructions of Rank Modulation Codes

We study error-correcting codes for permutations under the infinity norm, motivated the rank-modulation scheme for flash memories. In this scheme, a set of n flash cells are combined to create a single virtual multi-level cell. Information is stored in the permutation induced by the cell charge levels. Spike errors, which are characterized by a limited-magnitude change in cell charge levels, correspond to a low-distance change under the infinity norm. We define codes protecting against spike errors, called limited-magnitude rank-modulation codes (LMRM codes), and present several constructions for these codes, some resulting in optimal codes. These codes admit simple recursive, and sometimes direct, encoding and decoding procedures. We also provide lower and upper bounds on the maximal size of LMRM codes both in the general case, and in the case where the codes form a subgroup of the symmetric group. In the asymptotic analysis, the codes we construct out-perform the Gilbert-Varshamov-like bound estimate.

Correcting limited-magnitude errors in the rank-modulation scheme

Mappings of the set of binary vectors of a fixed length to the set of permutations of the same length are useful for the construction of permutation codes In this article, several explicit constructions of such mappings preserving or increasing the Hamming distance are given Some applications are given to illustrate the usefulness of the construction In particular, a new lower bound on the maximal size of permutation arrays (PAs) is given

/pdf/distance-preserving-mappings-from-binary-vectors-to-pe5dme8p1g.pdf

Distance-preserving mappings from binary vectors to permutations

We discuss the application of coded modulation for power line communications. We combine M-FSK with permutation codes to include frequency and time diversity. This makes the transmissions robust against permanent frequency disturbances and impulse noise. The scheme is applicable to any frequency range.

J. Haering

Papers

Coded M-FSK for power line communications