  Absolutely maximally entangled, AME, states are pure multipartite states that
give rise to the maximally mixed states when half or more of the parties are
traced out. AME states find applications in fields like teleportation or
quantum secret sharing, and the problem of finding conditions on their
existence has been considered in a number of papers.
  We consider here AME states of minimal support, that are simpler to analyse.
An equivalence with coding theory gives a sufficient condition for their
existence, that the number of sites be equal to the local dimension plus one,
when the local dimension $d$ is a power of a prime number.
  In this paper, an equivalence with latin hypercubes is used to prove that the
above sufficient condition fails in the first case in which the local dimension
is not a prime power, $d=6$. Results for other values of $d$ are also given.
