List Decoding for Oblivious Arbitrarily Varying MACs: Constrained and Gaussian

Abstract

This paper provides upper and lower bounds on list sizes of list decoding for two-user oblivious arbitrarily varying multiple access channels (AVMACs). An oblivious AVMAC consists of two users who wish to transmit messages (without cooperation) to a remote receiver, a malicious jammer who only has access to the codebooks of both users (which are also known to every party), and a receiver who is required to decode the message pair sent by both users. The transmitters send codewords which encode messages subject to input constraints. The jammer, without knowing the transmitted codeword pair, injects adversarial noise subject to state constraints so as to actively corrupt the communication from both users to the receiver. It was left as an open question by Cai (2016) to nail down the smallest list sizes for constrained AVMACs. Our inner and outer bounds are based on a judicious notion of symmetrizability for AVMACs introduced by Cai (2016) with twists to incorporate input and state constraints. The analysis follows techniques by Csisz\'ar and Narayan (1988). When no constraints are imposed, our bound collapse to prior results by Cai (2016) which characterized the list-decoding capacity region of unconstrained AVMACs. Techniques used in this paper can also be extended to the Gaussian case and we characterize the list-decoding capacity region for Gaussian AVMACs. The converse argument relies on a bounding technique recently used by Hosseinigoki and Kosut (2019).