Achievability Bounds of Coding with Finite Blocklength for Gaussian Broadcast Channels

In this paper, we study the achievable performance of dirty paper coding for the Gaussian broadcast channel (BC) with finite blocklength and we propose two different achievability bounds for this problem. We present the broadcast adaptation of dependence testing bound of Polyanskiy et al. 2010, which is an upper bound on the average error probability that depends on the channel dispersion terms of each error event for fixed input. Additionally, we introduce the $κβ$ lower bounds on the maximal code sizes of each user using dirty paper coding.

💡 Research Summary

The paper investigates the finite‑blocklength performance of dirty‑paper coding (DPC) over the two‑user Gaussian broadcast channel (GBC). While classical information theory characterizes the capacity region of the GBC in the asymptotic regime using superposition coding and successive cancellation, practical systems often operate with short packets, making it essential to understand the loss incurred by finite blocklengths. To this end, the authors adapt two non‑asymptotic achievability tools originally developed for single‑user additive white Gaussian noise (AWGN) channels—Polyanskiy’s dependence‑testing (DT) bound and the κβ bound—to the broadcast setting with DPC.

System model. The real‑valued GBC is described by Y_{j,i}=X_i+Z_{j,i}, i=1,…,n, j∈{1,2}, where Z_{j,i}∼N(0,N_j) are independent noises and N_2>N_1. The channel input is split as X_i = X_{1,i}+X_{2,i}, with X_{1,i}∼N(0,αP) and X_{2,i}∼N(0,(1−α)P) (α∈