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252, 378, 500, 634, 756 ], [ 125, 253, 379, 501, 635, 757 ], [ 126, 254, 380, 502, 636, 758 ], [ 127, 255, 381, 503, 637, 759 ] ] }, "distance": { "d": 6, "X": { "value": 6, "confidence": "upper_bound", "witness": [ 57, 193, 322, 441, 444, 689 ] }, "Z": { "value": 6, "confidence": "upper_bound", "witness": [ 82, 217, 345, 458, 466, 714 ] } }, "provenance": { "authors": [ "@mathysrennela" ], "construction": "Construction I of arXiv:2610.19451 (Nourozi, Smarandache, Mitchell, 'High-Girth QLDPC Codes from Circulant and Affine Permutation Matrices'), Eqs. (8)-(9), at N=128 (their Section V showcase): a fixed 2x6 circulant pattern H_X = [[I]*6; [C_a0..C_a5]], H_Z = [[C_b0..C_b5]; [C_c0..C_c5]] with shift lists a = (0,1,3,8,12,18), b = (0,-5,-5,0,12,12), c = (0,0,2,8,2,8) reduced mod N and C_s the NxN cyclic shift P_{t->t+s}. Theorem 1: CSS by even multiplicity of -b,-c,a-b,a-c; ranks 2N-1 per side; Tanner girth 12 for N=31 and every N >= 37; distance exactly 6, witnessed by the uniform weight-6 logical of their Eq. (12). The attached X-side witness is that operator mirrored through the incidence-graph isomorphism pi = (3,2,1,0,5,4) of the rank proof. Their Section V counts 3840 weight-6 logical supports per CSS side against 4864 for the comparable Komoto-Kasai arXiv:2501.13444 instance of the same [[768,258,6]].", "origin": "submission", "date": "2026-10-15", "model": "MiMo-V2.6-Flash", "notes": "Reproduction of arXiv:2610.19451 Construction I at N=128, the paper's own showcase instance, filed because it strictly beats the existing entry 960-258-6 on n (same k=258, d=6, w=6, 256 fewer qubits) and so advances the weight-6 x unrestricted frontier. 130 of the family's 131 admissible members are undominated; N=91 was not filed (parameters equal existing 546-184-6). Witness-backed upper bound d <= 6; the paper proves d = 6 exact. The [[n,k,d]] set matches Table I of the paper and the Komoto-Kasai arXiv:2501.13444 reference code.", "search_budget": { "candidates_screened": 135, "cpu_hours": 1.0, "tool": "numpy builder + gf2_fast RIS" } }, "family": "other" }