The following two papers will soon appear in the IEEE Transactions on Information Theory:
K. F. Trillingsgaard, W. Yang, G. Durisi, and P. Popovski, “Common-message broadcast channels with feedback in the nonasymptotic regime: Stop feedback,” IEEE Tran. Inf. Theory, 2018, to appear. [arXiv].
Abstract: We investigate the maximum coding rate for a given average blocklength and error probability over a $K$-user discrete memoryless broadcast channel for the scenario where a common message is transmitted using variable-length stop-feedback codes. For the point-to-point case, Polyanskiy et al. (2011) demonstrated that variable-length coding combined with stop-feedback significantly increases the speed of convergence of the maximum coding rate to capacity. This speed-up manifests itself in the absence of a square-root penalty in the asymptotic expansion of the maximum coding rate for large blocklengths, i.e., zero dispersion. In this paper, we present nonasymptotic achievability and converse bounds on the maximum coding rate of the common-message $K$-user discrete memoryless broadcast channel, which strengthen and generalize the ones reported in Trillingsgaard et al. (2015) for the two-user case. An asymptotic analysis of these bounds reveals that zero dispersion cannot be achieved for certain common-message broadcast channels (e.g., 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 expansions approximate accurately the maximum coding rate and that the speed of convergence to capacity is indeed slower than for the point-to-point case.
K. F. Trillingsgaard, W. Yang, G. Durisi, and P. Popovski, “Common-message broadcast channels with feedback in the nonasymptotic regime: Full feedback,” IEEE Tran. Inf. Theory, 2018, to appear [arXiv].
Abstract: We investigate the maximum coding rate achievable on a two-user broadcast channel for the case where a common message is transmitted with feedback using either ﬁxed-blocklength codes or variable-length codes. For the ﬁxed-blocklength-code setup, we establish nonasymptotic converse and achievability bounds. An asymptotic analysis of these bounds reveals that feedback improves the second-order term compared to the no-feedback case. In particular, for a certain class of antisymmetric broadcast channels, we show that the dispersion is halved. For the variable-length-code setup, we demonstrate that the channel dispersion is zero.