On Interference Networks over Finite Fields

We present a framework to study linear deterministic interference networks over finite fields. Unlike the popular linear deterministic models introduced to study Gaussian networks, we consider networks where the channel coefficients are general scalars over some extension field $\FF_{p^m}$ (scalar $m$-th extension-field models), $m \times m$ diagonal matrices over $\FF_p$ ($m$-symbol extension ground-field models), and $m \times m$ general non-singular matrices (MIMO ground field models). We use the companion matrix representation of the extension field to convert $m$-th extension scalar models into MIMO ground-field models where the channel matrices have special algebraic structure. For such models, we consider the $2 \times 2 \times 2$ topology (two-hops two-flow) and the 3-user interference network topology. We derive achievability results and feasibility conditions for certain schemes based on the Precoding-Based Network Alignment (PBNA) approach, where intermediate nodes use random linear network coding (i.e., propagate random linear combinations of their incoming messages) and non-trivial precoding/decoding is performed only at the network edges, at the sources and destinations. Furthermore, we apply this approach to the scalar $2\times 2\times 2$ complex Gaussian IC with fixed channel coefficients, and show two competitive schemes outperforming other known approaches at any SNR, where we combine finite-field linear precoding/decoding with lattice coding and the Compute and Forward approach at the signal level. As a side result, we also show significant advantages of vector linear network coding both in terms of feasibility probability (with random coding coefficients) and in terms of coding latency, with respect to standard scalar linear network coding, in PBNA schemes.