  Schalkwijk and Kailath (1966) developed a class of block codes for Gaussian
channels with ideal feedback for which the probability of decoding error
decreases as a second-order exponent in block length for rates below capacity.
This well-known but surprising result is explained and simply derived here in
terms of a result by Elias (1956) concerning the minimum mean-square distortion
achievable in transmitting a single Gaussian random variable over multiple uses
of the same Gaussian channel. A simple modification of the Schalkwijk-Kailath
scheme is then shown to have an error probability that decreases with an
exponential order which is linearly increasing with block length. In the
infinite bandwidth limit, this scheme produces zero error probability using
bounded expected energy at all rates below capacity. A lower bound on error
probability for the finite bandwidth case is then derived in which the error
probability decreases with an exponential order which is linearly increasing in
block length at the same rate as the upper bound.
