  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.
