Optimality of Simple Layered Superposition Coding in the 3 User MISO BC with Finite Precision CSIT

We study the $K=3$ user multiple input single output (MISO) broadcast channel (BC) with $M=3$ antennas at the transmitter and $1$ antenna at each receiver, from the generalized degrees of freedom (GDoF) perspective, under the assumption that the channel state information at the transmitter (CSIT) is limited to finite precision. In particular, our goal is to identify a parameter regime where a simple layered superposition (SLS) coding scheme achieves the entire GDoF region. With $\alpha_{ij}$ representing the channel strength parameter for the link from the $j^{th}$ antenna of the transmitter to the $i^{th}$ receiver, we prove that SLS is GDoF optimal without the need for time-sharing if $\max(\alpha_{ki},\alpha_{im})\leq\alpha_{ii}$ and $\alpha_{ki}+\alpha_{im}\le\alpha_{ii}+\alpha_{km}$ for all $i,k\in[3],m\in[M]$. The GDoF region under this condition is a convex polyhedron. The result generalizes to arbitrary $M\geq 3$.