Optimal finite-length linear codes and the corresponding channel dispersion for broadcast packet erasure channels with feedback

With the recent emergence of many low-latency applications over wireless networks, the need for accurate finite-length analysis of channel coding over multi-user wireless channels is ever increasing. This paper focuses exclusively on the two-user broadcast packet erasure channel (PEC) with causal feedback, for which existing results show that various linear network coding (LNC) schemes can attain the broadcast capacity region when the block length approaches infinity. Instead of the asymptotic capacity-based analysis, this work derives the exact value of the LNC-based broadcast channel dispersion. Our approach is based on a new explicit characterization of the optimal LNC scheme under any arbitrarily given finite block length. The results show that among all existing asymptotically capacity-achieving LNC schemes, one (class) of them is provably finite-length optimal. By analyzing its second-order asymptotic, we have thus derived the exact (optimal) LNC broadcast channel dispersion, which closes the gap of the state-of-the-art inner and outer bounds previously derived in Lin et al. ISIT 2021.