On Base Field of Linear Network Coding

For a (single-source) multicast network, the size of a base field is the most known and studied algebraic identity that is involved in characterizing its linear solvability over the base field. In this paper, we design a new class N of multicast networks and obtain an explicit formula for the linear solvability of these networks, which involves the associated coset numbers of a multiplicative subgroup in a base field. The concise formula turns out to be the first that matches the topological structure of a multicast network and algebraic identities of a field other than size. It further facilitates us to unveil infinitely many new multicast networks linearly solvable over GF(q) but not over GF(q') with q <; q', based on a subgroup order criterion. In particular: 1) for every k ≥ 2, an instance in N can be found linearly solvable over GF(2^2k) but not over GF(2^2k+1) and 2) for arbitrary distinct primes p and p', there are infinitely many k and k' such that an instance in N can be found linearly solvable over GF(p^k) but not over GF(p'^k') with p^k <; p'^k'.