Unifying notions of generalized weights for universal security on wire-tap networks
Unifying notions of generalized weights for universal security on wire-tap networks
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DOI：
10.1109/allerton.2016.7852315
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发表时间：
2016-07
期刊：
影响因子：
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通讯作者：
Umberto Martínez-Peñas;R. Matsumoto
中科院分区：
文献类型：
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作者：
Umberto Martínez-Peñas;R. Matsumoto
对线性网络编码网络的普遍安全性进行了深入研究。然而，以前用于此目的的线性码在比在网络上使用的更大的域上是线性的。在这项工作中，我们为在网络中使用的域上线性的线性码引入了新的参数(相对维度/秩支撑度和相对广义矩阵重量)，以衡量这些码的普遍安全性能。与以前的工作不同，所提出的新参数使我们能够在无噪声网络上对所有可能的参数使用最优的通用安全线性码，并且还使我们能够在Guruswami等人最近提出的列表可译码秩度量码的基础上增加通用安全性。我们给出了新参数的几个性质：单调性，Singleton型上下界，对偶定理，以及线性码等价的定义和刻画。最后，我们证明了我们的参数严格推广了相对维长分布和相对广义Hamming权，以及相对维交分布和相对广义秩权。此外，我们还证明了广义矩阵权大于Delsarte广义权。
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 optimally universal secure linear codes on noiseless networks 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 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. Moreover, we show that generalized matrix weights are larger than Delsarte generalized weights.