  In this paper, we consider a communication system where a sender sends
messages over a memoryless Gaussian point-to-point channel to a receiver and
receives the output feedback over another Gaussian channel with known variance
and unit delay. The sender sequentially transmits the message over multiple
times till a certain error performance is achieved. The aim of our work is to
design a transmission strategy to process every transmission with the
information that was received in the previous feedback and send a signal so
that the estimation error drops as quickly as possible. The optimal code is
unknown for channels with noisy output feedback when the block length is
finite. Even within the family of linear codes, optimal codes are unknown in
general. Bridging this gap, we propose a family of linear sequential codes and
provide a dynamic programming algorithm to solve for a closed form expression
for the optimal code within a class of sequential linear codes. The optimal
code discovered via dynamic programming is a generalized version of which the
Schalkwijk-Kailath (SK) scheme is one special case with noiseless feedback; our
proposed code coincides with the celebrated SK scheme for noiseless feedback
settings.
