This talk 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 the order of $q^{(\ell-1)t^2/\ell}-q^t$,

for any $q \geq 2$, where $t$ denotes the dimension of the vector solution,

and the number of messages is $2 \ell$, ${\ell \geq 2}$. 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. Similar results are given for any number of odd messages greater

than two. The results are obtained by considering

several multicast networks which are variations of the well-known combination network.

Joint work with Tuvi Etzion.