On vector linear solvability of multicast networks

In the literature of network coding, vector linear network coding (LNC) is a generalization of the conventional scalar LNC, such that the data unit transmitted on every edge is an L-dimensional vector of data symbols over a base field GF(q). A scalar linear code over GF(q) is simply a vector linear code of dimension 1 over GF(q), and a general network has a scalar linear solution over GF(q^L) only if it has a vector linear solution of dimension L over GF(q). Though vector LNC is more powerful in enabling a higher coding diversity, this work will present explicit multicast networks, for the first time in the literature, with the special property that they do not have a vector linear solution of dimension L over GF(2) but have scalar linear solutions over GF(q′), for some q′ < 2^L. This reveals the fact that although vector LNC can outperform scalar LNC in terms of yielding a solution for a general network, scalar LNC can also outperform vector LNC of dimension larger than 1 in terms of using a smaller alphabet to yield a solution for a multicast network.