  A symmetric butterfly network (BFN) with a full-duplex relay operating in a
bi-directional fashion for feedback is considered. This network is relevant for
a variety of wireless networks, including cellular systems dealing with
cell-edge users. Upper bounds on the capacity region of the general memoryless
BFN with feedback are derived based on cut-set and cooperation arguments and
then specialized to the linear deterministic BFN with really-source feedback.
It is shown that the upper bounds are achievable using combinations of the
compute-forward strategy and the classical decode-and-forward strategy, thus
fully characterizing the capacity region. It is shown that net rate gains are
possible in certain parameter regimes.
