The Dispersion of Broadcast Channels With Degraded Message Sets Using Spherical Codebooks

We study the two-user broadcast channel with degraded message sets and derive second-order achievability rate regions. Specifically, the channel noises are not necessarily Gaussian and we use spherical codebooks for both users. The weak user with worse channel quality applies nearest neighbor decoding by treating the signal of the other user as interference. For the strong user with better channel quality, we consider two decoding schemes: successive interference cancellation (SIC) decoding and joint nearest neighbor (JNN) decoding. We adopt two performance criteria: separate error probabilities (SEP) and joint error probability (JEP). Under our analysis, SIC and JNN decoding share the same second-order achievable rate region despite the fact that JNN decoding often yields better performance in other multiterminal problems. Furthermore, we generalize our results to the case with quasi-static fading and show that the asymptotic notion of outage capacity region is an accurate performance measure even at finite blocklengths.