On network coding with finite channel state information

We study network coding for wireless networks with finite channel state information (CSI) at intermediate nodes (relays). Based on the CSI of the relay-sink channels, we adapt the network codes at the relays. For a specific network with two sources, four relays and two sinks, the analytic results show that one bit CSI of all relay-sink channels (global CSI) can reduce complexity (field size), and simultaneously decrease the erasure probability. Then, we generalize the results to relay networks with M users, N relays and J sinks. We show that fixed network codes without CSI cannot achieve instantaneous min-cut, i.e., min-cut under current channel state. We also show that with one bit global CSI, we can achieve instantaneous min-cut by adapting the network codes using an alphabet size L, where L is the number of sinks connecting to a relay. Yet, the fixed MDS network codes use an alphabet size L(_M-1^N-1). For the networks with perfect or imperfect source-relay channels, adaptive network codes with one bit global CSI have lower erasure probability than the codes without CSI. Thus, one bit global CSI can reduce the erasure probability, and simultaneously reduce coding complexity.