  The problem of transmitting a parameter value over an additive white Gaussian
noise (AWGN) channel is considered, where, in addition to the transmitter and
the receiver, there is a helper that observes the noise non-causally and
provides a description of limited rate $R_h$ to the transmitter and/or the
receiver. We derive upper and lower bounds on the optimal achievable
$\alpha$-th moment of the estimation error and show that they coincide for
small values of $\alpha$ and for high values of $R_h$. The upper bound relies
on a recently proposed channel-coding scheme that effectively conveys $R_h$
bits essentially error-free and the rest of the rate - over the same AWGN
channel without help, with the error-free bits being allocated to the most
significant bits of the quantized parameter. We then concentrate on the setting
with a total transmit energy constraint, for which we derive achievability
results for both channel coding and parameter modulation for several scenarios:
when the helper assists only the transmitter or only the receiver and knows the
noise, and when the helper assists the transmitter and/or the receiver and
knows both the noise and the message. In particular, for the message-informed
helper that assists both the receiver and the transmitter, it is shown that the
error probability in the channel-coding task decays doubly exponentially.
Finally, we translate these results to those for continuous-time power-limited
AWGN channels with unconstrained bandwidth. As a byproduct, we show that the
capacity with a message-informed helper that is available only at the
transmitter can exceed the sum of the capacity without help and the help rate
$R_h$, when the helper knows only the noise but not the message.
