  This work considers the smart repeater network where a single source $s$
wants to send two independent packet streams to destinations $\{d_1,d_2\}$ with
the help of relay $r$. The transmission from $s$ or $r$ is modeled by packet
erasure channels: For each time slot, a packet transmitted by $s$ may be
received, with some probabilities, by a random subset of $\{d_1,d_2,r\}$; and
those transmitted by $r$ will be received by a random subset of $\{d_1,d_2\}$.
Interference is avoided by allowing at most one of $\{s,r\}$ to transmit in
each time slot. One example of this model is any cellular network that supports
two cell-edge users when a relay in the middle uses the same downlink resources
for throughput/safety enhancement.
  In this setting, we study the capacity region of $(R_1,R_2)$ when allowing
linear network coding (LNC). The proposed LNC inner bound introduces more
advanced packing-mixing operations other than the previously well-known
butterfly-style XOR operation on overheard packets of two co-existing flows. A
new LNC outer bound is derived by exploring the inherent algebraic structure of
the LNC problem. Numerical results show that, with more than 85% of the
experiments, the relative sum-rate gap between the proposed outer and inner
bounds is smaller than 0.08% under the strong-relaying setting and 0.04% under
arbitrary distributions, thus effectively bracketing the LNC capacity of the
smart repeater problem.
