  The optimal coding scheme for communicating a Gaussian message over an
Additive White Gaussian noise (AWGN) channel with AWGN output feedback, with a
limited number of transmissions is unknown. Even if we restrict the scope of
the coding scheme to linear schemes, still, deriving the optimal coding scheme
is a challenging task. The state-of-the-art linear scheme for channels with
noisy feedback is by Chance and Love, where the coefficients of the linear
scheme are numerically optimized based on unique observations [1]. In this
paper, we introduce a new class of sequential linear schemes for this channel
by introducing a novel linear state process at the transmitter and derive the
optimal sequential scheme within this class of schemes in a closed-form by
formulating a novel Dynamic Programming (DP). We empirically show that our
scheme outperforms the state-of-the-art linear scheme in [1] for noisy feedback
and coincides with the SK scheme for noiseless feedback. We also show that in
communicating message bits as opposed to a Gaussian message, a learning-based
approach further improves the reliability of sequential linear schemes. This
problem is an instance of decentralized control without any common information
and to the best of our knowledge the first such scenario where we can derive
analytical solutions using a DP.
