New Inner and Outer Bounds for 2-User Gaussian Broadcast Channels with Heterogeneous Blocklength Constraints

We investigate both a novel inner and outer bound on the rate region of a 2-user Gaussian broadcast channel with finite, heterogeneous blocklength constraints (HB-GBC). In particular, we introduce a new, modified Sato-type outer bound that can be applied in the finite blocklength regime and does not require the same marginal property. We then develop and analyze concatenated shell codes, which are suitable for the HB-GBC. Especially, to achieve a smaller decoding latency for the user with shorter blocklength constraint when successive interference cancellation is used, we derive the number of symbols needed to successfully early decode the other user's message. We numerically compare our derived outer bound to the best known achievable rate regions. Numerical results show that the new early decoding performance is significantly improved compared to the state of the art, and performs very close to the asymptotic limit.