  There has been substantial progress recently in understanding toy problems of
purely implicit signaling. These are problems where the source and the channel
are implicit -- the message is generated endogenously by the system, and the
plant itself is used as a channel. In this paper, we explore how implicit and
explicit communication can be used synergistically to reduce control costs. The
setting is an extension of Witsenhausen's counterexample where a rate-limited
external channel connects the two controllers. Using a semi-deterministic
version of the problem, we arrive at a binning-based strategy that can
outperform the best known strategies by an arbitrarily large factor. We also
show that our binning-based strategy attains within a constant factor of the
optimal cost for an asymptotically infinite-length version of the problem
uniformly over all problem parameters and all rates on the external channel.
For the scalar case, although our results yield approximate optimality for each
fixed rate, we are unable to prove approximately-optimality uniformly over all
rates.
