A [[58,21,9]] Binary Quantum Stabilizer Code: Improving the Best Known Minimum Distance for [[58,21]] from 8 to 9

This deposit presents an explicit binary quantum stabilizer code with parameters [[58,21,9]]2, improving the best known minimum distance for length 58, dimension 21 from d = 8 to d = 9. The previous entry in the tables of bounds on quantum codes maintained by Markus Grassl (codetables.de) had been unchanged since 2006-04-03. Construction. The code is a stabilizer extension of the stored [[58,28,8]] code from the same tables: seven mutually commuting elements of the parent code's normalizer are adjoined to its stabilizer, chosen such that every one of the parent's 8004 minimum-weight (weight-8) logical operators anticommutes with the extended stabilizer. The extension elements were found by simulated annealing combined with exact GF(2) linear solving over the parent's 56-dimensional logical space. Proof of d = 9. Two logically independent, exhaustive computations in exact integer arithmetic establish d ≥ 9: (i) a cover argument — the parent normalizer contains no nonzero words of symplectic weight ≤ 7 and exactly 8004 words of weight 8 (both established by exhaustive meet-in-the-middle enumeration), and all 8004 anticommute with the extended stabilizer, while the child normalizer is contained in the parent normalizer; (ii) a direct exhaustive sweep of the child code's normalizer at weights 1–8, finding zero words. Three explicit weight-9 logical operators establish d ≤ 9. Independent verification. An adversarial verification prompt (included verbatim) was executed twice by an independent AI system (ChatGPT, OpenAI) with code execution. Both runs independently confirmed the structural claims and the witness, audited the cover argument's logic, and ran falsification searches (including a complete enumeration of all 35,214,034 weight-≤4 words and a 20-partition meet-in-the-middle search with ≈99.9% per-counterexample detection probability) without finding any word of weight ≤ 8. Two additional weight-9 logical operators found by the verifier were re-verified by the authors' engine and are included. The verifier correctly noted that the exact d ≥ 9 floor rests on the authors' exhaustive sweeps, corroborated probabilistically by its own searches. Contents. Generator matrix of the new code (37 rows, explicit [X|Z] form, conventions documented); the parent [[58,28,8]] matrix; three weight-9 witnesses; the complete list of the parent's 8004 weight-8 logical words; the seven extension elements; the full machine-readable certificate; the complete verification toolchain (Python 3 / numpy, all decisive predicates in exact integer arithmetic); and the adversarial verification prompt as executed. Methodology note. The target cell was identified by a systematic analysis of derivation chains in the code tables: the [[58,21,8]] entry was a subcode-propagation shadow of the [[58,28,8]] parent, one of a family of such entries at lengths 55–65 untouched since 2006. As a documented negative result, the single-position distance-preserving puncture of Gundersen et al. (2025) provably does not exist for this parent: all 58 positions exhaust all three nonzero symplectic values over the 8004 minimum-weight words. This work was carried out as a human–AI collaboration (AI assistance: Claude, Anthropic; independent adversarial verification: ChatGPT, OpenAI); all decisive claims rest on exact, reproducible computation rather than on assertions of the AI systems. This deposit precedes disclosure to [e-mail].