This paper investigates the problem of single-source multicasting over a communication network in the presence of restricted adversaries. When the adversary is constrained to operate only on a prescribed subset of edges, classical cut-set bounds are no longer tight, and achieving capacity typically requires a joint design of the outer code and the inner (network) code. This stands in sharp contrast with the case of unrestricted adversaries, where capacity can be achieved by combining linear network coding with appropriate rank-metric outer codes. Building on the framework of network decoding, we determine the exact one-shot capacity of one of the fundamental families of 2-level networks introduced in [4], and we improve the best currently known lower bounds for another such family. In addition, we introduce a new family of networks that generalizes several known examples, and derive partial capacity results that illustrate a variety of phenomena that arise specifically in the restricted-adversary setting. Finally, we investigate the concept of separability of networks with respect to both the rank metric and the Hamming metric.

翻译：本文研究了在受限对手存在的情况下，通过通信网络进行单源多播的问题。当对手被限制仅能在预设的边子集上操作时，经典的割集界不再紧致，实现容量通常需要外层码与内层（网络）码的联合设计。这与无限制对手的情况形成鲜明对比，后者可以通过将线性网络编码与适当的秩度量外层码相结合来实现容量。基于网络解码框架，我们确定了文献[4]中引入的基本2层网络家族之一的精确单次容量，并针对另一个此类家族改进了当前已知的最佳下界。此外，我们引入了一个新的网络家族，它推广了多个已知示例，并推导了部分容量结果，这些结果说明了在受限对手设定下特有的多种现象。最后，我们研究了网络在秩度量与汉明度量下的可分性概念。