Low-capacity scenarios have become increasingly important in the technology of the Internet of Things (IoT) and the next generation of wireless networks. Such scenarios require efficient and reliable transmission over channels with an extremely small capacity. Within these constraints, the state-of-the-art coding techniques may not be directly applicable. Moreover, the prior work on the finite-length analysis of optimal channel coding provides inaccurate predictions of the limits in the low-capacity regime. In this paper, we study channel coding at low capacity from two perspectives: fundamental limits at finite length and code constructions. We first specify what a low-capacity regime means. We then characterize finite-length fundamental limits of channel coding in the low-capacity regime for various types of channels, including binary erasure channels (BECs), binary symmetric channels (BSCs), and additive white Gaussian noise (AWGN) channels. From the code construction perspective, we characterize the optimal number of repetitions for transmission over binary memoryless symmetric (BMS) channels, in terms of the code blocklength and the underlying channel capacity, such that the capacity loss due to the repetition is negligible. Furthermore, it is shown that capacity-achieving polar codes naturally adopt the aforementioned optimal number of repetitions.

翻译：低容量场景在物联网（IoT）和下一代无线网络技术中日益重要。此类场景要求信道在容量极小的条件下实现高效可靠的传输。在此约束下，现有先进编码技术可能无法直接适用。此外，现有关于最优信道编码有限长分析的工作在低容量区域提供了不精确的极限预测。本文从有限长基本极限和码构造两个角度研究低容量条件下的信道编码。我们首先界定低容量区域的范畴，随后针对包括二进制删除信道（BEC）、二进制对称信道（BSC）和加性高斯白噪声（AWGN）信道在内的多种信道类型，刻画低容量区域下信道编码的有限长基本极限。从码构造角度，我们基于码块长和底层信道容量，刻画了在二进制无记忆对称（BMS）信道上传输时最优重复次数的特征，使得由重复引起的容量损失可忽略。此外，研究表明，可达容量的极化码自然采用了上述最优重复次数。