122 research outputs found

    It'll probably work out: improved list-decoding through random operations

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    In this work, we introduce a framework to study the effect of random operations on the combinatorial list-decodability of a code. The operations we consider correspond to row and column operations on the matrix obtained from the code by stacking the codewords together as columns. This captures many natural transformations on codes, such as puncturing, folding, and taking subcodes; we show that many such operations can improve the list-decoding properties of a code. There are two main points to this. First, our goal is to advance our (combinatorial) understanding of list-decodability, by understanding what structure (or lack thereof) is necessary to obtain it. Second, we use our more general results to obtain a few interesting corollaries for list decoding: (1) We show the existence of binary codes that are combinatorially list-decodable from 1/2ϵ1/2-\epsilon fraction of errors with optimal rate Ω(ϵ2)\Omega(\epsilon^2) that can be encoded in linear time. (2) We show that any code with Ω(1)\Omega(1) relative distance, when randomly folded, is combinatorially list-decodable 1ϵ1-\epsilon fraction of errors with high probability. This formalizes the intuition for why the folding operation has been successful in obtaining codes with optimal list decoding parameters; previously, all arguments used algebraic methods and worked only with specific codes. (3) We show that any code which is list-decodable with suboptimal list sizes has many subcodes which have near-optimal list sizes, while retaining the error correcting capabilities of the original code. This generalizes recent results where subspace evasive sets have been used to reduce list sizes of codes that achieve list decoding capacity

    On the List-Decodability of Random Linear Rank-Metric Codes

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    The list-decodability of random linear rank-metric codes is shown to match that of random rank-metric codes. Specifically, an Fq\mathbb{F}_q-linear rank-metric code over Fqm×n\mathbb{F}_q^{m \times n} of rate R=(1ρ)(1nmρ)εR = (1-\rho)(1-\frac{n}{m}\rho)-\varepsilon is shown to be (with high probability) list-decodable up to fractional radius ρ(0,1)\rho \in (0,1) with lists of size at most Cρ,qε\frac{C_{\rho,q}}{\varepsilon}, where Cρ,qC_{\rho,q} is a constant depending only on ρ\rho and qq. This matches the bound for random rank-metric codes (up to constant factors). The proof adapts the approach of Guruswami, H\aa stad, Kopparty (STOC 2010), who established a similar result for the Hamming metric case, to the rank-metric setting

    A Combinatorial Bound on the List Size

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    In this paper we study the scenario in which a server sends dynamic data over a single broadcast channel to a number of passive clients. We consider the data to consist of discrete packets, where each update is sent in a separate packet. On demand, each client listens to the channel in order to obtain the most recent data packet. Such scenarios arise in many practical applications such as the distribution of weather and traffic updates to wireless mobile devices and broadcasting stock price information over the Internet. To satisfy a request, a client must listen to at least one packet from beginning to end. We thus consider the design of a broadcast schedule which minimizes the time that passes between a clients request and the time that it hears a new data packet, i.e., the waiting time of the client. Previous studies have addressed this objective, assuming that client requests are distributed uniformly over time. However, in the general setting, the clients behavior is difficult to predict and might not be known to the server. In this work we consider the design of universal schedules that guarantee a short waiting time for any possible client behavior. We define the model of dynamic broadcasting in the universal setting, and prove various results regarding the waiting time achievable in this framework

    Bounds on List Decoding of Rank-Metric Codes

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    So far, there is no polynomial-time list decoding algorithm (beyond half the minimum distance) for Gabidulin codes. These codes can be seen as the rank-metric equivalent of Reed--Solomon codes. In this paper, we provide bounds on the list size of rank-metric codes in order to understand whether polynomial-time list decoding is possible or whether it works only with exponential time complexity. Three bounds on the list size are proven. The first one is a lower exponential bound for Gabidulin codes and shows that for these codes no polynomial-time list decoding beyond the Johnson radius exists. Second, an exponential upper bound is derived, which holds for any rank-metric code of length nn and minimum rank distance dd. The third bound proves that there exists a rank-metric code over \Fqm of length nmn \leq m such that the list size is exponential in the length for any radius greater than half the minimum rank distance. This implies that there cannot exist a polynomial upper bound depending only on nn and dd similar to the Johnson bound in Hamming metric. All three rank-metric bounds reveal significant differences to bounds for codes in Hamming metric.Comment: 10 pages, 2 figures, submitted to IEEE Transactions on Information Theory, short version presented at ISIT 201
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