Symmetry-guided constructions of absolutely maximally entangled states in five open cases

Abstract

We give explicit Hermitian self-dual maximum distance separable codes with parameters $[12,6,7{]}_{25}$, $[18,9,10{]}_{121}$, and $[18,9,10{]}_{169}$. The stabilizer construction proves the existence of $\mathrm{AME}(12,5)$, $\mathrm{AME}(18,11)$, and $\mathrm{AME}(18,13)$ states; one-party projection also gives $\mathrm{AME}(17,11)$ and $\mathrm{AME}(17,13)$. The first code was found by a direct search. A regular ${\mathbb{Z}}_3^2$ coordinate orbit of its automorphism group suggested a group-circulant form that reduces each length-eighteen construction to a nine-entry kernel. The printed matrices are certified by exact Hermitian products and complete square-minor enumeration.

4 pages. Results were found and written up with the help of Claude Fable 5 (Anthropic) and ChatGPT 5.6 Sol (OpenAI)