On base field of linear network coding

A few (single-source) multicast networks were recently discovered with the special property of linearly solvable over a finite field GF(q) but not over a larger GF(q′). In this paper, these networks are extended to a general class N of multicast networks. We obtain a concise condition, in terms of multiplicative subgroup orders in GF(q), for networks in N to be linearly solvable over GF(q). This full characterization facilitates us to design infinitely many new multicast networks linearly solvable over GF(q) but not over GF(q′) with q < q′, based on a subgroup order criterion. As an interesting instance among them, a network linearly solvable over GF(22k) but not over GF(22k+1), can be constructed for every k ⩾ 2. Our findings suggest that the suitability of a field for a given network depends on not only the size and the characteristic of the field, but also the matching between the algebraic structure of the field and the topological structure of the network.