  We consider dense coding with partially entangled states on bipartite systems
of dimension $d\times d$, studying the conditions under which a given number of
messages, $N$, can be deterministically transmitted. It is known that the
largest Schmidt coefficient, $\lambda_0$, must obey the bound $\lambda_0\le
d/N$, and considerable empirical evidence points to the conclusion that there
exist states satisfying $\lambda_0=d/N$ for every $d$ and $N$ except the
special cases $N=d+1$ and $N=d^2-1$. We provide additional conditions under
which this bound cannot be reached -- that is, when it must be that
$\lambda_0<d/N$ -- yielding insight into the shapes of boundaries separating
entangled states that allow $N$ messages from those that allow only $N-1$. We
also show that these conclusions hold no matter what operations are used for
the encoding, and in so doing, identify circumstances under which unitary
encoding is strictly better than non-unitary.
