  Pure multiparticle quantum states are called absolutely maximally entangled
if all reduced states obtained by tracing out at least half of the particles
are maximally mixed. We provide a method to characterize these states for a
general multiparticle system. With that, we prove that a seven-qubit state
whose three-body marginals are all maximally mixed, or equivalently, a pure
$((7,1,4))_2$ quantum error correcting code, does not exist. Furthermore, we
obtain an upper limit on the possible number of maximally mixed three-body
marginals and identify the state saturating the bound. This solves the
seven-particle problem as the last open case concerning maximally entangled
states of qubits.
