Exact Minimum Distance of a ⟦𝟖𝟑𝟐, 𝟏𝟎⟧ Quantum LDPC Code: 𝐝 = 𝟏𝟓 by Exact Fold-and-Lift Enumeration

We determine the exact minimum distance of a two-block quasi-cyclic CSS quantum LDPC code with n = 832 physical qubits and k = 10 logical qubits (genus5 GB CLEAN, QECTOR artifact lineage). Earlier releases had established 10 ≤ d ≤ 15. We prove d = 15: in each CSS sector the classical kernel contains no nonzero word of weight 1 to 14 and exactly 416 words of weight 15, forming a single orbit under simultaneous cyclic shifts, all of them nontrivial logical operators. The proof uses the repeated-root structure of the kernel over F2[x]/(x^416 + 1): since 416 = 13·2^5 and x^(2m) + 1 = (x^m + 1)^2 over F2, the kernel folds exactly to a 26-bit code and lifts back without loss, giving an exact, symmetry-reduced enumeration of low-weight words. An 18-check validation suite passes, a separately written re-implementation reproduces every intermediate count, and both implementations show that no kernel word has weight 16 to 20. Computer-assisted proof, not formally verified. Version 2.1 supersedes version 1.0 (version 2.0 was an internal revision, not released). Changes: summary box, notation table, algorithm listing and pipeline figure; narrower novelty statement, related-work section and 20 additional references, independent re-implementation, extended enumeration to weight 20, full re-run for bundle release 1.2.2, corrected availability statement. Results unchanged. The code specification is published in Appendix C, and the verification bundle, with machine-checked proofs of the upper bound, is public (doi:10.5281/zenodo.23463892).