Hermitian self-dual MDS codes for absolutely maximally entangled states: exact counts and monomial classification
Abstract
A Hermitian self-dual MDS code [2k,k,k+1]_(q²) produces, through the stabilizer construction, an absolutely maximally entangled state on 2k parties of local dimension q, and such codes are exactly the k × k matrices A over 𝔽_(q²) with Aoverline(A)^(T)=-Iₖ all of whose square submatrices are nonsingular. We determine the exact number of these codes in nineteen cells, including 30 569 011 200 codes [8,4,5]₂₅, 2 642 411 520 codes [10,5,6]₉, and none at all in the cell [8,4,5]₉; at k=2 the count is (q-2)(q+1)³ for every q we reach. We classify thirteen cells up to monomial equivalence: the [8,4,5]₂₅ cell has exactly seven classes, the [10,5,6]₉ cell exactly one, and the [6,3,4]_(q²) cells have 1,1,4,8,30,51 classes for q=2,3,5,7,11,13. Finally we answer, for the [12,6,7]₂₅ code constructed by Bevins and Bidav (arXiv:2608.05781), the equivalence question their paper leaves open: its monomial automorphism group has order 108, its class contains exactly 9 654 465 016 627 200 codes, and its Galois image is a second, inequivalent class, so the classification of that cell is nontrivial. Every numerical statement below is machine-checked in Lean 4.
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Archived files
- Version 1 · current (opens in a new tab)
Source snapshot 2026-08-30 15:34 UTC
File fingerprint
af35aba207b53842402a9b8665a14d7acfa7bd8cd423c8106901c613105b2e19
Claim ledger
Stated results
ame-01known data2026-08-23
[12,6,7]₂5 Hermitian self-dual MDS: the AME(12,5) certificate of arXiv:2608.05781
ame-02known data2026-08-23
[18,9,10]₁21 from the Z₃² group-circulant kernel: the AME(18,11) / AME(17,11) certificate
ame-03known data2026-08-23
[18,9,10]₁69: the AME(18,13) / AME(17,13) certificate
ame-04routine2026-08-23
no linear [8,4,5]₄ MDS code: AME(8,4) has no linear minimal-support realisation
ame-05routine2026-08-23
[6,3,4]₄ hexacode and the 486 superregular 3x3 blocks over F₄: the AME(6,2) gate
ame-06routine2026-08-23
[6,3,4]₉ over F₉ and no linear [6,3,4]₃: AME(6,3) exists but has no minimal-support linear realisation
ame-07routine2026-08-23
[8,4,5]₂5 over F₂5: the AME(8,5) gate
ame-08routine2026-08-23
the five field models are fields, re-verified over the whole field
ame-09routine2026-08-23
the memoised Plucker table equals the Laplace determinant on all 923 minors
ame-10routine2026-08-23
negative controls: Hermitian-but-not-MDS, MDS-but-not-Hermitian, a corrupted A5, and a corrupted kernel
ame-11routine2026-08-23
no F₁6-linear Hermitian self-dual MDS [8,4,5]₁6 code: the arXiv:2608.05781 route cannot reach AME(8,4)
ame-12known2026-08-29
no F₄-linear stabilizer AME(8,4): the additive class is empty, by exhaustion over M₂(F₄) block matrices
ame-13known2026-08-29
Bush's bound in Mathlib: an (n,qᵏ,n-k+1)_q MDS code has n <= q+k-1, so a minimal-support AME(2k,q) needs k <= q-1 and there is no (8,256,5)₄ code
ame-14routine2026-08-29
the additive search's controls: the k=2,3 searches return AME(4,4) and AME(6,4) and each half of the predicate is separately satisfiable at k=4
ame-15routine2026-08-29
M₂(F₄) as a self-checking block model, the SL(2,4) coset facts that license the search's normal form, and the invariance identities its soundness consumes
ame-16candidate2026-08-30
exact counts of Hermitian self-dual MDS [2k,k,k+1]_(q²) codes for 15 cells: the (q−2)(q+1)³ law at k=2, [8,4,5]₂5 = 30 569 011 200, [10,5,6]₉ = 2 642 411 520, [8,4,5]₉ = 0
ame-17routine2026-08-30
the reduction the counts stand on: free diagonal action, transversal, unreduced re-runs, and the k=6 level-1 gate
ame-18candidate2026-08-30
the monomial class of the Bevins–Bidav [12,6,7]₂5 code: |Aut| = 108, class size 9 654 465 016 627 200, and its Galois image as a second, inequivalent class
ame-19candidate2026-08-30
classification up to monomial equivalence of thirteen cells: [8,4,5]₂5 has exactly 7 classes, [10,5,6]₉ exactly 1, [6,3,4]_(49/121/169) have 8/30/51
ame-20routine2026-08-30
the equivalence machinery's controls: hexacode |Aut| = 1080 = |3·A₆| and Danielsen's [10,5,6]₉ uniqueness recovered; kernel ranks dim C^(⋆2) = 12 (the paper's own invariant) and dim(C ⋆ conj C) = 2k−1
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- An AME(n, q) state — n q-level systems, every reduction to at most ⌊n/2⌋ parties maximally mixed — is the same thing as a pure quantum MDS code [[n, 0, ⌊n/2⌋+1]]_q. Whether one exists is, in general, a question about a unit vector in (C^q)^⊗n: real-algebraic, not finite.
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
801848d7