Variable-Length Coding with Stop-Feedback for the Common-Message Broadcast Channel

This paper investigates the maximum coding rate over a K-user discrete memoryless broadcast channel for the scenario where a common message is transmitted using variable-length stop-feedback codes. Specifically, upon decoding the common message, each decoder sends a stop signal to the encoder, which transmits continuously until it receives all K stop signals. We present nonasymptotic achievability and converse bounds for the maximum coding rate, which strengthen and generalize the bounds previously reported in Trillingsgaard et al. (2015) for the two-user case. An asymptotic analysis of these bounds reveal that---contrary to the point-to-point case---the second-order term in the asymptotic expansion of the maximum coding rate decays inversely proportional to the square root of the average blocklength. This holds for certain nontrivial common-message broadcast channels, such as the binary symmetric broadcast channel. Furthermore, we identify conditions under which our converse and achievability bounds are tight up to the second order. Through numerical evaluations, we illustrate that our second-order asymptotic expansion approximates accurately the maximum coding rate and that the speed of convergence to capacity is indeed slower than for the point-to-point case.