  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.
