What was already known
The q=5 and q=11 cases were open in the Huber-Wyderka table, while its table did not cover q=13. Earlier Hermitian self-dual codes at length 12 over F25 reached distance 6 rather than 7, and at length 18 over F121/F169 reached distance 9 rather than 10. The paper says the q=13 parameters were unresolved in the sources it reviewed.
AI involvement
Bevins and Bidav directed the project and independently reviewed the constructions and calculations. Claude Fable 5 and ChatGPT 5.6 Sol performed the computational search; the paper does not assign individual matrices to one model.
Model-run search produced the central retained constructions within a human-directed research project.
Importance
Gives exact code certificates for five previously unresolved AME existence parameters.
Limitations
The certificates establish existence in the five stated cases, not a classification of AME states. The q=13 cases were unresolved in the sources the authors reviewed, while the earlier comparison table did not cover q=13. Independent checking beyond the authors is not documented.
Technical verification
The authors report complete exact finite-field checks of the printed matrices and independently reviewed those computations. The checks cover Hermitian self-duality and every square minor, which, with the stated standard coding criterion, establish the five existence claims. These checks are source-reported.
Formal checking
Finite-field Hermitian products and Gaussian-elimination enumeration of all 923 minors for the 6-by-6 matrix and 48,619 minors for each 9-by-9 matrix
- Checker
- Bevins and Bidav's exact-arithmetic computation
- Scope
- The three printed generator matrices are Hermitian self-dual and MDS
- Basis
- Reported by the linked checker
Independent human review of the constructions, computations, and full text
- Checker
- Samuel Bevins and Yunus Bidav, authors
- Scope
- The three code constructions and the resulting five AME existence statements
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The authors explain the systematic MDS criterion, the orbit that motivated group-circulant kernels, and the exact certificate. No central mechanism is identified as unexplained.
Correction and publication history
Gives exact code certificates for five previously unresolved AME existence parameters.
Read this published revision