  In this paper we prove that no absolutely maximally entangled, AME, state
with minimal support exists with 7 sites and 5 levels.
  General AME states are pure multipartite states that, when reduced to half or
less of the sites, the maximum entropy mixed state is obtained. They have found
applications in teleportation and quantum secret sharing, and finding
conditions for their existence is a well known open problem. We consider the
version of this problem for minimally supported AME states. We single out known
both sufficient and necessary conditions in that case. From our negative
result, we show that the necessary condition is not sufficient. The proof uses
a recent result on the theory of general, nonlinear, classical codes.
