Reliability of the Gaussian broadcast channel with common message and feedback

We consider the two-user memoryless Gaussian broadcast channel (BC) with feedback and common message only. We show that linear-feedback schemes with message points, in the spirit of Schalkwijk&Kailath's scheme for point-to-point channels or Ozarow&Leung's scheme for BCs with private messages, are strictly suboptimal for this setup. In fact even with perfect feedback, the maximum rate achieved by these schemes is strictly smaller than capacity (which is the same with and without feedback). In contrast, rate-limited feedback suffices for bursty-feedback schemes to achieve double-exponential decay of the probability of error when the feedback rate R
FB
is at least equal to the forward rate R.