12,022 research outputs found

    Correcting Charge-Constrained Errors in the Rank-Modulation Scheme

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    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 τ-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 τ-metric. Specifically, the one-error-correcting codes we construct are at least half the ball-packing upper bound

    Systematic Error-Correcting Codes for Rank Modulation

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    The rank-modulation scheme has been recently proposed for efficiently storing data in nonvolatile memories. Error-correcting codes are essential for rank modulation, however, existing results have been limited. In this work we explore a new approach, \emph{systematic error-correcting codes for rank modulation}. Systematic codes have the benefits of enabling efficient information retrieval and potentially supporting more efficient encoding and decoding procedures. We study systematic codes for rank modulation under Kendall's τ\tau-metric as well as under the \ell_\infty-metric. In Kendall's τ\tau-metric we present [k+2,k,3][k+2,k,3]-systematic codes for correcting one error, which have optimal rates, unless systematic perfect codes exist. We also study the design of multi-error-correcting codes, and provide two explicit constructions, one resulting in [n+1,k+1,2t+2][n+1,k+1,2t+2] systematic codes with redundancy at most 2t+12t+1. We use non-constructive arguments to show the existence of [n,k,nk][n,k,n-k]-systematic codes for general parameters. Furthermore, we prove that for rank modulation, systematic codes achieve the same capacity as general error-correcting codes. Finally, in the \ell_\infty-metric we construct two [n,k,d][n,k,d] systematic multi-error-correcting codes, the first for the case of d=O(1)d=O(1), and the second for d=Θ(n)d=\Theta(n). In the latter case, the codes have the same asymptotic rate as the best codes currently known in this metric

    New Bounds for Permutation Codes in Ulam Metric

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    New bounds on the cardinality of permutation codes equipped with the Ulam distance are presented. First, an integer-programming upper bound is derived, which improves on the Singleton-type upper bound in the literature for some lengths. Second, several probabilistic lower bounds are developed, which improve on the known lower bounds for large minimum distances. The results of a computer search for permutation codes are also presented.Comment: To be presented at ISIT 2015, 5 page

    Generalized Gray Codes for Local Rank Modulation

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    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 rank-modulation 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.Comment: 7 pages, 1 figure, shorter version was submitted to ISIT 201

    Space-Time Signal Design for Multilevel Polar Coding in Slow Fading Broadcast Channels

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    Slow fading broadcast channels can model a wide range of applications in wireless networks. Due to delay requirements and the unavailability of the channel state information at the transmitter (CSIT), these channels for many applications are non-ergodic. The appropriate measure for designing signals in non-ergodic channels is the outage probability. In this paper, we provide a method to optimize STBCs based on the outage probability at moderate SNRs. Multilevel polar coded-modulation is a new class of coded-modulation techniques that benefits from low complexity decoders and simple rate matching. In this paper, we derive the outage optimality condition for multistage decoding and propose a rule for determining component code rates. We also derive an upper bound on the outage probability of STBCs for designing the set-partitioning-based labelling. Finally, due to the optimality of the outage-minimized STBCs for long codes, we introduce a novel method for the joint optimization of short-to-moderate length polar codes and STBCs

    Constant-Weight Gray Codes for Local Rank Modulation

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    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. The local rank-modulation, as a generalization of the rank-modulation scheme, has been recently suggested as a way of storing information in flash memory. We study constant-weight Gray codes for the local rank-modulation scheme in order to simulate conventional multi-level flash cells while retaining the benefits of rank modulation. We provide necessary conditions for the existence of cyclic and cyclic optimal Gray codes. We then specifically study codes of weight 2 and upper bound their efficiency, thus proving that there are no such asymptotically-optimal cyclic codes. In contrast, we study codes of weight 3 and efficiently construct codes which are asymptotically-optimal
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