15,913 research outputs found

    Estimates on the Size of Symbol Weight Codes

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    The study of codes for powerlines communication has garnered much interest over the past decade. Various types of codes such as permutation codes, frequency permutation arrays, and constant composition codes have been proposed over the years. In this work we study a type of code called the bounded symbol weight codes which was first introduced by Versfeld et al. in 2005, and a related family of codes that we term constant symbol weight codes. We provide new upper and lower bounds on the size of bounded symbol weight and constant symbol weight codes. We also give direct and recursive constructions of codes for certain parameters.Comment: 14 pages, 4 figure

    Coding for Errors and Erasures in Random Network Coding

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    The problem of error-control in random linear network coding is considered. A ``noncoherent'' or ``channel oblivious'' model is assumed where neither transmitter nor receiver is assumed to have knowledge of the channel transfer characteristic. Motivated by the property that linear network coding is vector-space preserving, information transmission is modelled as the injection into the network of a basis for a vector space VV and the collection by the receiver of a basis for a vector space UU. A metric on the projective geometry associated with the packet space is introduced, and it is shown that a minimum distance decoder for this metric achieves correct decoding if the dimension of the space V∩UV \cap U is sufficiently large. If the dimension of each codeword is restricted to a fixed integer, the code forms a subset of a finite-field Grassmannian, or, equivalently, a subset of the vertices of the corresponding Grassmann graph. Sphere-packing and sphere-covering bounds as well as a generalization of the Singleton bound are provided for such codes. Finally, a Reed-Solomon-like code construction, related to Gabidulin's construction of maximum rank-distance codes, is described and a Sudan-style ``list-1'' minimum distance decoding algorithm is provided.Comment: This revised paper contains some minor changes and clarification

    A Rank-Metric Approach to Error Control in Random Network Coding

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    The problem of error control in random linear network coding is addressed from a matrix perspective that is closely related to the subspace perspective of K\"otter and Kschischang. A large class of constant-dimension subspace codes is investigated. It is shown that codes in this class can be easily constructed from rank-metric codes, while preserving their distance properties. Moreover, it is shown that minimum distance decoding of such subspace codes can be reformulated as a generalized decoding problem for rank-metric codes where partial information about the error is available. This partial information may be in the form of erasures (knowledge of an error location but not its value) and deviations (knowledge of an error value but not its location). Taking erasures and deviations into account (when they occur) strictly increases the error correction capability of a code: if μ\mu erasures and δ\delta deviations occur, then errors of rank tt can always be corrected provided that 2t≤d−1+μ+δ2t \leq d - 1 + \mu + \delta, where dd is the minimum rank distance of the code. For Gabidulin codes, an important family of maximum rank distance codes, an efficient decoding algorithm is proposed that can properly exploit erasures and deviations. In a network coding application where nn packets of length MM over FqF_q are transmitted, the complexity of the decoding algorithm is given by O(dM)O(dM) operations in an extension field FqnF_{q^n}.Comment: Minor corrections; 42 pages, to be published at the IEEE Transactions on Information Theor

    Product Construction of Affine Codes

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    Binary matrix codes with restricted row and column weights are a desirable method of coded modulation for power line communication. In this work, we construct such matrix codes that are obtained as products of affine codes - cosets of binary linear codes. Additionally, the constructions have the property that they are systematic. Subsequently, we generalize our construction to irregular product of affine codes, where the component codes are affine codes of different rates.Comment: 13 pages, to appear in SIAM Journal on Discrete Mathematic

    Generalized List Decoding

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    This paper concerns itself with the question of list decoding for general adversarial channels, e.g., bit-flip (XOR\textsf{XOR}) channels, erasure channels, AND\textsf{AND} (ZZ-) channels, OR\textsf{OR} channels, real adder channels, noisy typewriter channels, etc. We precisely characterize when exponential-sized (or positive rate) (L−1)(L-1)-list decodable codes (where the list size LL is a universal constant) exist for such channels. Our criterion asserts that: "For any given general adversarial channel, it is possible to construct positive rate (L−1)(L-1)-list decodable codes if and only if the set of completely positive tensors of order-LL with admissible marginals is not entirely contained in the order-LL confusability set associated to the channel." The sufficiency is shown via random code construction (combined with expurgation or time-sharing). The necessity is shown by 1. extracting equicoupled subcodes (generalization of equidistant code) from any large code sequence using hypergraph Ramsey's theorem, and 2. significantly extending the classic Plotkin bound in coding theory to list decoding for general channels using duality between the completely positive tensor cone and the copositive tensor cone. In the proof, we also obtain a new fact regarding asymmetry of joint distributions, which be may of independent interest. Other results include 1. List decoding capacity with asymptotically large LL for general adversarial channels; 2. A tight list size bound for most constant composition codes (generalization of constant weight codes); 3. Rederivation and demystification of Blinovsky's [Bli86] characterization of the list decoding Plotkin points (threshold at which large codes are impossible); 4. Evaluation of general bounds ([WBBJ]) for unique decoding in the error correction code setting
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