Precoding based network Alignment and the capacity of a finite field X channel

Precoding based network alignment (PBNA) is a network coding paradigm inspired by wireless networks where all the intelligence resides at the sources and destination nodes whereas intermediate relay nodes only perform arbitrary linear network coding operations, creating an effective one-hop finite field linear network between sources and destinations. The main question explored in this work is how degrees of freedom (DoF) results from wireless networks can be translated into capacity results for their finite field counterparts. A finite field X channel, i.e., a multiple unicast network comprised of 2 source nodes, 2 destination nodes and 4 independent messages (one for each source-destination pair) is considered in this work, where the channel outputs are arbitrary linear combinations of channel inputs over a finite field F_pn. Like its wireless counterpart, which has 4/3 sum DoF, this channel is shown to have a sum capacity of 4/3 symbols per channel use for most channel realizations. The main insight is that, with a few exceptions that are pointed out, scalar (SISO) finite field channels over F_pn are analogous to n×n complex vector (MIMO) channels in the wireless setting, so the DoF optimal precoding solutions for wireless networks can be translated into capacity optimal solutions for their finite field counterparts.