Multicast Network Coding and Field Sizes

In an acyclic multicast network, it is well known that a linear network coding solution over GF(q) exists when q is sufficiently large. In particular, for each prime power q no smaller than the number of receivers, a linear solution over GF(q) can be efficiently constructed. In this paper, we reveal that a linear solution over a given finite field does not necessarily imply the existence of a linear solution over all larger finite fields. In particular, we prove by construction that: 1) for every ω ≥ 3, there is a multicast network with source outdegree ω linearly solvable over GF(7) but not over GF(8), and another multicast network linearly solvable over GF(16) but not over GF(17); 2) there is a multicast network linearly solvable over GF(5) but not over such GF(q) that q > 5 is a Mersenne prime plus 1, which can be extremely large; 3) a multicast network linearly solvable over GF(q^m1) and over GF(q^m2) is not necessarily linearly solvable over GF(q^m1+m2); and 4) there exists a class of multicast networks with a set T of receivers such that the minimum field size q_min for a linear solution over GF(q_min) is lower bounded by O(√|T|), but not every larger field than GF(q_min) suffices to yield a linear solution. The insight brought from this paper is that not only the field size but also the order of subgroups in the multiplicative group of a finite field affects the linear solvability of a multicast network.