We study linear encoding for a pair of correlated
Gaussian sources transmitted over a two-user Gaussian broadcast
channel in the presence of unit-delay noiseless feedback, abbre-
viated as the GBCF. Each pair of source samples is transmitted
using a linear transmission scheme in a
finite
number of channel
uses. We investigate three linear transmission schemes: A scheme
based on the Ozarow-Leung (OL) code, a scheme based on
the linear quadratic Gaussian (LQG) code of Ardestanizadeh
et al., and a novel scheme derived in this work using a dynamic
programming (DP) approach. For the OL and LQG schemes
we present lower and upper bounds on the minimal number of
channel uses needed to achieve a target mean-square error (MSE)
pair. For the LQG scheme in the symmetric setting, we identify
the optimal scaling of the sources, which results in a significant
improvement of its finite horizon performance, and, in addition,
characterize the (exact) minimal number of channel uses required
to achieve a target MSE. Finally, for the symmetric setting, we
show that for any fixed and finite number of channel uses, the
DP scheme achieves an MSE lower than the MSE achieved by
either the LQG or the OL schemes.