On Base Field of Linear Network Coding - Sun, Qifu Tyler; Li, Shuo-Yen Robert; Zongpeng Li (IEEE transactions on information theory, 2016)
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' .