Vector network coding based on subspace codes outperforms scalar linear network coding

This paper considers vector network coding based on rank-metric codes and subspace codes. Our main result is that vector network coding can significantly reduce the required field size compared to scalar linear network coding in the same multicast network. The achieved gap between the field size of scalar and vector network coding is in q^{(h-2)t2/h+o(t)} for any q ≥ 2 and any even h ≥ 4, where t denotes the dimension of the vector solution and h the number of messages. If h ≥ 5 is odd, then the achieved gap of the field size between the scalar network coding solution and the vector network coding solution is q^{(h-3)t2/(h-1)+o(t)}. Previously, only a gap of constant size had been shown. This implies also the same gap between the field size in linear and non-linear scalar network coding for multicast networks. The results are obtained by considering several multicast networks which are variations of the well-known combination network.