Gaussian Multiple and Random Access in the Finite Blocklength Regime

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 \(\frac12\frac{\log n}{n}+O \left(\frac 1 n \right)\) bits per channel use. The result 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.