Symmetry-guided construction and exact certification of absolutely maximally entangled states

We construct Hermitian self-dual maximum-distance-separable codes with parameters $[12,6,7]_{25}$, $[18,9,10]_{121}$, and $[18,9,10]_{169}$. Through the nonbinary stabilizer construction these codes define $\text{AME}(12,5)$, $\text{AME}(18,11)$, and $AMD(18,13)$ states, and one-party projection gives $\text{AME}(17,11)$ and $\text{AME}(17,13)$. The $[12,6,7]_{25}$ code, which is not monomially equivalent to a generalized Reed-Solomon code, is obtained by a systematic search without prescribed coordinate symmetry; its monomial-semilinear automorphism group acts with a regular $\mathbb{Z}_3^2$ orbit on nine coordinates. Imposing that action on two nine-coordinate orbits reduces the length-eighteen search to a nine-element group algebra kernel, whose Hermitian character decomposition splits the invariant self-duality constraints into independent blocks with exact family counts; a norm-one coordinate scaling removes a factor $q+1=14$ at $q=13$. Hermitian self-duality and the MDS property are certified by exact field arithmetic and complete square-minor enumeration. Graph-state cut-rank computations, including all single-vertex deletions of the length-eighteen graphs, provide additional exact checks.