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    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
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