  Absolutely maximally entangled (AME) states are pure multi-partite
generalizations of the bipartite maximally entangled states with the property
that all reduced states of at most half the system size are in the maximally
mixed state. AME states are of interest for multipartite teleportation and
quantum secret sharing and have recently found new applications in the context
of high-energy physics in toy models realizing the AdS/CFT-correspondence. We
work out in detail the connection between AME states of minimal support and
classical maximum distance separable (MDS) error correcting codes and, in
particular, provide explicit closed form expressions for AME states of $n$
parties with local dimension $q$ a power of a prime for all $q \geq n-1$.
Building on this, we construct a generalization of the Bell-basis consisting of
AME states and develop a stabilizer formalism for AME states. For every $q \geq
n-1$ prime we show how to construct QECCs that encode a logical qudit into a
subspace spanned by AME states. Under a conjecture for which we provide
numerical evidence, this construction produces a family of quantum error
correcting codes $[\![n,1,n/2]\!]_q$ for $n$ even, saturating the quantum
Singleton bound. We show that our conjecture is equivalent to the existence of
an operator whose support cannot be decreased by multiplying it with stabilizer
products and explicitly construct the codes up to $n = 8$.
