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Unifying notions of generalized weights for universal security on wire-tap networks

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arXiv ID: 1607.01263
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Universal security over a network with linear network coding has been intensively studied. However, previous linear codes used for this purpose were linear over a larger field than that used on the network. In this work, we introduce new parameters (relative dimension/rank support profile and relative generalized matrix weights) for linear codes that are linear over the field used in the network, measuring the universal security performance of these codes. The proposed new parameters enable us to use maximum rank distance linear codes for all possible parameters, as opposed to previous works, and also enable us to add universal security to the recently proposed list-decodable rank-metric codes by Guruswami et al. We give several properties of the new parameters: monotonicity, Singleton-type lower and upper bounds, a duality theorem, and definitions and characterizations of equivalences and degenerateness of linear codes. Finally, we show that our parameters strictly extend relative dimension/length profile and relative generalized Hamming weights, respectively, and relative dimension/intersection profile and relative generalized rank weights, respectively. The duality theorems for generalized Hamming weights and generalized rank weights can be deduced as special cases of the proposed duality theorem for generalized matrix weights. Moreover, generalized matrix weights are larger than Delsarte generalized weights.


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