A random coding approach to Gaussian multiple access channels with finite blocklength

Contrary to the common use of random coding and typicality decoding for the achievability proofs in information theory, the tightest achievable rates for point-to-point Gaussian channels build either on geometric arguments or composite hypothesis testing, for which direct generalization to multi-user settings appears challenging. In this paper, we provide a new perspective on the procedure of handling input cost constraints for tight achievability results. In particular, we show with a proper choice of input distribution and using a change of measure technique, tight bounds can be achieved via the common random coding argument and a modified typicality decoding. It is observed that a codebook generated randomly according to a uniform distribution on the “power shell” is optimal, at least up to the second order. Such insights are then extended to a Gaussian multiple access channel, for which independent uniform distributions on power shells are shown to be very close to optimal, at least up to second order.