On finite-length analysis and channel dispersion for broadcast packet erasure channels with feedback

Motivated by the applications for low-delay communication networks, the finite-length analysis, or channel dispersion identification, of the multi-user channel is very important. Recent studies also incorporate the effects of feedback in point-to-point and common-message broadcast channels (BCs). However, with private messages and feedback, finite-length results for BCs are much more scarce. Though it is known that feedback can strictly enlarge the capacity, the ultimate feedback capacity regions remain unknown for even some classical channels including Gaussian BCs. In this work, we study the two-user broadcast packet erasure channel (PEC) with causal feedback, which is one of the cleanest feedback capacity results and the capacity region can be achieved by elegant linear network coding (LNC). We first derive a new finite-length outer bound for any LNCs and then accompanying inner bound by analyzing a three-phase LNC. For the outer-bound, we adopt a linear-space-based framework, which can successfully find the LNC capacity. However, naively applying this method in finite-length regime will result in a loose outer bound. Thus a new bounding technique based on carefully labelling each time slot according to the type of LNC transmitted is proposed. Simulation results show that the sum-rate gap between our inner and outer bounds is within 0.02 bits/channel use. Asymptotic analysis also shows that our bounds bracket the channel dispersion of LNC feedback capacity for broadcast PEC to within a factor of Q-l (E/2)/Q-l (E).