Gaussian Multiple and Random Access Channels: Finite-Blocklength Analysis

This paper presents finite-blocklength achievability bounds for the Gaussian multiple access channel (MAC) and random access channel (RAC) under average-error and maximal-power constraints. Using random codewords uniformly distributed on a sphere and a maximum likelihood decoder, the derived MAC bound on each transmitter's rate matches the MolavianJazi-Laneman bound (2015) in its first- and second-order terms, improving the remaining terms to \frac {1}2\frac {\log {n}}{n}+{O} \left ({\frac {1}{n}}\right) bits per channel use. The result \vphantom {\sum ^{R}} then extends to a RAC model in which neither the encoders nor the decoder knows which of {K} possible transmitters are active. In the proposed rateless coding strategy, decoding occurs at a time {n}_{t} that depends on the decoder's estimate {t} of the number of active transmitters {k} . Single-bit feedback from the decoder to all encoders at each potential decoding time {n}_{i} , {i} \leq {t} , informs the encoders when to stop transmitting. For this RAC model, the proposed code achieves the same first-, second-, and third-order performance as the best known result for the Gaussian MAC in operation.