On the Power of Vector Linear Network Coding

This paper presents yet another instance of the power of vector linear network coding over scalar linear network coding. Previous works have established that the size of the finite field required to achieve a vector linear solution may be smaller than that size of the finite field required to achieve a scalar linear solution. It has been also shown there exist networks which do not have a scalar linear solution but have a vector linear solution. In this paper we show that the set of characteristics over which a network has a vector linear solution may be larger than the set of characteristics over which it has a scalar linear solution. We prove this result by showing a network which has a scalar linear solution if and only if the characteristic of the finite field is 2, but has a 2-dimensional vector linear solution over every finite fields.