  The energy-optimal scheme is found for communicating one bit over a
memoryless Gaussian channel with an ideal feedback channel. It is assumed that
the channel is allowed to be used at most N times before decoding. The optimal
coding/decoding strategy is derived by dynamic programming. It is found that
feedback gives a significant performance gain and that the optimal strategies
are discontinuous. It is also shown that most of the performance increase can
be obtained even with a one-bit feedback channel. The optimal scheme is
compared with the strategy by Kailath-Schalkwijk and is found to be
significantly more effective. For the case of a diagonal MIMO channel where
measurement noise variances are equal along the sub channels we also show that
the problem can be reduced to the previous case of transmitting one bit over a
scalar feedback channel.
