  Deterministic linear network coding (DLNC) is an important family of network
coding techniques for wireless packet broadcast. In this paper, we show that
DLNC is strongly related to and can be effectively studied using matroid theory
without bridging index coding. We prove the equivalence between the DLNC
solution and matrix matroid. We use this equivalence to study the performance
limits of DLNC in terms of the number of transmissions and its dependence on
the finite field size. Specifically, we derive the sufficient and necessary
condition for the existence of perfect DLNC solutions and prove that such
solutions may not exist over certain finite fields. We then show that
identifying perfect solutions over any finite field is still an open problem in
general. To fill this gap, we develop a heuristic algorithm which employs
graphic matroids to find perfect DLNC solutions over any finite field.
Numerical results show that its performance in terms of minimum number of
transmissions is close to the lower bound, and is better than random linear
network coding when the field size is not so large.
