  A pure multipartite quantum state is called absolutely maximally entangled
(AME), if all reductions obtained by tracing out at least half of its parties
are maximally mixed. Maximal entanglement is then present across every
bipartition. The existence of such states is in many cases unclear. With the
help of the weight enumerator machinery known from quantum error correction and
the generalized shadow inequalities, we obtain new bounds on the existence of
AME states in dimensions larger than two. To complete the treatment on the
weight enumerator machinery, the quantum MacWilliams identity is derived in the
Bloch representation. Finally, we consider AME states whose subsystems have
different local dimensions, and present an example for a $2 \times3 \times 3
\times 3$ system that shows maximal entanglement across every bipartition.
