An achievable rate region for the multiple-access channel with feedback

An achievable rate region
R_{1} \leq I(X_{1};Y|X_{2},U), R_{2} \leq I(X_{2}; Y|X_{1},U), R_{1}+R_{2} \leq I(X_{1}, X_{2};Y)
, where
p(u,x_{l},x_{2},y)= p(u)p(x_{l}|u)p(x_{2}|u)p(y|x_{l},x_{2})
, is established for the multiple-access channel with feedback. Time sharing of these achievable rates yields the rate region of this paper. This region generally exceeds the achievable rate region without feedback and exceeds the rate point found by Gaarder and Wolf for the binary erasure multiple-access channel with feedback. The presence of feedback allows the independent transmitters to understand each other's intended transmissions before the receiver has sufficient information to achieve the desired decoding. This allows the transmitters to cooperate in the transmission of information that resolves the residual uncertainty of the receiver. At the same time, independent information from the transmitters is superimposed on the cooperative correction information. The proof involves list codes and block Markov encoding.