Absolutely Maximally Entangled States in Five Open Cases

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.

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.