标题： 统一无线窃听网络上通用安全性的广义权重概念 显示英文标题

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

摘要： 在网络中使用线性网络编码的通用安全性已被深入研究。 然而，之前用于此目的的线性码是在比网络中使用的域更大的域上进行的。 在本工作中，我们引入了新的参数（相对维度/秩支持分布和相对广义矩阵权重）用于在网络中使用的域上进行的线性码，以衡量这些码的通用安全性性能。 所提出的新的参数使我们能够在无噪声网络中为所有可能的参数最优地使用通用安全线性码，与之前的工作相反，并且还使我们能够通过Guruswami等人最近提出的可列表解码的秩度量码添加通用安全性。 我们给出了新参数的几个性质：单调性、Singleton型下界和上界、对偶定理，以及线性码等价性的定义和特征。 最后，我们表明我们的参数分别严格扩展了相对维度/长度分布和相对广义汉明权重，以及相对维度/交集分布和相对广义秩权重。 此外，我们表明广义矩阵权重大于Delsarte广义权重。 显示英文摘要

摘要： 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.