The synchronization problem in network having line, star or combined linestar (tree) topology via communication links with limited capacity is considered. Limit possibilities of nonlinear controlled synchronization systems with firstorder coders for multi-dimensional master-slave Lurie systems (linear part plus nonlinearity depending only on measurable outputs) are presented. It is shown that the upper and lower bounds of limited synchronization error are proportional to the maximum coupling signal rate and inversely proportional to the minimum information transmission rate of network communication links (minimum channel capacity). The results are illustrated by example of synchronization in the network of Chua systems coupled via communication links with limited capacity.