基于群代数的网络码置换与相加
Permute & Add Network Codes via Group Algebras
摘要：
在文献中已经提出了一类网络码，其中在网络边缘上传输的符号是二进制向量，并且在网络节点中执行的编码操作包括对每个输入向量应用（可能是几个）置换并对结果进行XOR以获得输出向量。这些网络码，我们将称为“置换和相加”网络码，涉及更简单的操作，并且已知提供比标量线性网络码更低复杂度的解决方案。这些代码的复杂度由它们的“度”决定，“度”是应用于每个输入向量以计算输出向量的置换的数量。用于多播网络的置换和添加网络码的构造是已知的。在本文中，我们提供了一个新的框架，基于群代数的任意（不一定是组播）网络设计置换和添加网络代码。我们的框架允许使用任何有限组的排列（包括循环移位，在以前的作品中提出），并承认编码率和代码的程度之间的权衡。此外，我们的技术允许优雅的恢复和概括的关键结果在文献中已知的置换和添加网络代码。
英文摘要：
A class of network codes have been proposed in the literature where the symbols transmitted on network edges are binary vectors and the coding operation performed in network nodes consists of the application of (possibly several) permutations on each incoming vector and XOR-ing the results to obtain the outgoing vector. These network codes, which we will refer to as "permute-and-add" network codes, involve simpler operations and are known to provide lower complexity solutions than scalar linear network codes. The complexity of these codes is determined by their "degree" which is the number of permutations applied on each incoming vector to compute an outgoing vector. Constructions of permute-and-add network codes for multicast networks are known. In this paper, we provide a new framework based on group algebras to design permute-and-add network codes for arbitrary (not necessarily multicast) networks. Our framework allows the use of any finite group of permutations (including circular shifts, proposed in prior works) and admits a trade-off between coding rate and the degree of the code. Further, our technique permits elegant recovery and generalizations of the key results on permute-and-add network codes known in the literature.
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