  We investigate the entanglement properties in the symmetric subspace of
$N$-partite $d$-dimensional systems (qudits). For diagonal symmetric states, we
show that there is no bound entanglement for $d = 3,4 $ and $N = 3$. Further,
we present a constructive algorithm to map multipartite diagonal symmetric
states of qudits onto bipartite symmetric states of larger local dimension.
This technique greatly simplifies the analysis of multipartite states and
allows to infer entanglement properties for any even $N \geq 4 $ due to the
fact that the PPT conditions that arise from the bipartite symmetric state
correspond to the same PPT conditions that appear in the multipartite diagonal
symmetric state.
