  We propose a novel multipath multi-hop adaptive and causal random linear
network coding (AC-RLNC) algorithm with forward error correction. This
algorithm generalizes our joint optimization coding solution for point-to-point
communication with delayed feedback. AC-RLNC is adaptive to the estimated
channel condition, and is causal, as the coding adjusts the retransmission
rates using a priori and posteriori algorithms. In the multipath network, to
achieve the desired throughput and delay, we propose to incorporate an adaptive
packet allocation algorithm for retransmission, across the available resources
of the paths. This approach is based on a discrete water filling algorithm,
i.e., bit-filling, but, with two desired objectives, maximize throughput and
minimize the delay. In the multipath multi-hop setting, we propose a new
decentralized balancing optimization algorithm. This balancing algorithm
minimizes the throughput degradation, caused by the variations in the channel
quality of the paths at each hop. Furthermore, to increase the efficiency, in
terms of the desired objectives, we propose a new selective recoding method at
the intermediate nodes. We derive bounds on the throughput and the mean and
maximum in order delivery delay of AC-RLNC, both in the multipath and multipath
multi-hop case. In the multipath case, we prove that in the non-asymptotic
regime, the suggested code may achieve more than 90% of the channel capacity
with zero error probability. In the multipath multi-hop case, the balancing
procedure is proven to be optimal with regards to the achieved rate. Through
simulations, we demonstrate that the performance of our adaptive and causal
approach, compared to selective repeat (SR)-ARQ protocol, is capable of gains
up to a factor two in throughput and a factor of more than three in delay.
