This paper introduces a network linear block coding framework for multi-way relaying with differential MPSK modulation. We consider a system with $K$ user terminals and $L$ relays employing a selective detect-and-forward protocol. We show that such a system can be represented as a systematic $(K+L,K)$ linear block code. This framework allows the use of linear systematic block codes as network codes yielding power saving gains at user terminals. We analyse the theoretical performance of our scheme with optimal decoding, showing that our system can achieve a diversity order that equals to the minimum Hamming distance of the associated code. Then we consider a sub-optimal decoder based on log-domain belief propagation, and present numerical results for three 4-ary linear block codes and two binary LDPC codes as examples. Theoretical and simulation results demonstrate a significant performance gain of our scheme over uncoded transmission, even though some relays may be silent occasionally. We present a technique based on hard thresholding the received samples at the terminals, that makes the performance closer to the case when no relay is silent, without any significant increase in complexity. Furthermore, we also present guidelines for choosing suitable codes, and show that systematic block codes with sparse generator matrices are in particular desirable for our system.

中文翻译：

---

一种差分调制的选择性检测转发多路中继的网络线性块编码方法

本文介绍了一种用于具有差分 MPSK 调制的多路中继的网络线性块编码框架。我们考虑一个系统 $K$ 用户终端和 $L$ 中继采用选择性检测转发协议。我们表明，这样的系统可以表示为一个系统的 $(K+L,K)$ 线性块码。该框架允许使用线性系统分组码作为在用户终端产生节能增益的网络码。我们通过最佳解码分析了我们方案的理论性能，表明我们的系统可以实现等于相关代码的最小汉明距离的分集阶数。然后我们考虑基于对数域置信传播的次优解码器，并给出三个 4 进制线性块码和两个二进制 LDPC 码的数值结果作为例子。理论和仿真结果表明，我们的方案比未编码传输具有显着的性能提升，即使某些中继可能偶尔会静音。我们提出了一种基于对终端接收到的样本进行硬阈值处理的技术，这使得性能更接近没有中继静默的情况，没有任何显着增加复杂性。此外，我们还提出了选择合适代码的指南，并表明具有稀疏生成器矩阵的系统块代码特别适合我们的系统。