Absolutely Maximally Entangled States in Five Open Cases

A pure state of $n$ parties with $q$ levels each is absolutely maximally entangled, written $\mathrm{AME}(n,q)$, when every subsystem of at most $\lfloor n/2 \rfloor$ parties is maximally mixed. These are the perfect tensors, and existence is a parameter-by-parameter problem: some $(n,q)$ admit one, some provably do not, and a maintained table records which cells are still unknown. This paper settles five of them. It exhibits Hermitian self-dual MDS codes $[12,6,7]_{25}$, $[18,9,10]_{121}$ and $[18,9,10]_{169}$, from which the stabilizer construction gives $\mathrm{AME}(12,5)$, $\mathrm{AME}(18,11)$ and $\mathrm{AME}(18,13)$, and projecting one party gives $\mathrm{AME}(17,11)$ and $\mathrm{AME}(17,13)$.

Five existence statements, all by explicit construction: $\mathrm{AME}(12,5)$, $\mathrm{AME}(17,11)$, $\mathrm{AME}(18,11)$, $\mathrm{AME}(17,13)$ and $\mathrm{AME}(18,13)$. The $[12,6,7]_{25}$ code came from a direct search with no symmetry imposed; its automorphism group turned out to have a regular $\mathbb{Z}_3^2$ coordinate orbit, and imposing that translation symmetry on two nine-coordinate orbits collapses an unrestricted $9 \times 9$ block to a nine-element kernel, which is what made the length-eighteen searches feasible. The symmetry is search scaffolding, not part of the proof: the three printed matrices and the two checks suffice on their own. The paper is explicit that the searches were not exhaustive, so it proves existence and classifies nothing - equivalence and classification for these parameters stay open. The length-twelve code is also shown not to be monomially equivalent to a generalized Reed-Solomon code.