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[ "@MicheldeMessieres" ], "date": "2026-10-14", "found_at_samples": 69893, "survived_samples": 2000000, "seeds": [ 1011 ], "tool": "Stern ISD (own C++), numpy-checked; survived 2M board gf2_fast RIS" } }, "Z": { "value": 22, "confidence": "upper_bound", "witness": [ 54, 99, 168, 228, 251, 279, 302, 325, 374, 434, 439, 485, 493, 542, 668, 730, 744, 767, 880, 908, 910, 922 ], "witness_provenance": { "found_by": [ "@MicheldeMessieres" ], "date": "2026-10-14", "found_at_samples": 77047, "survived_samples": 2000000, "seeds": [ 1111 ], "tool": "Stern ISD (own C++), numpy-checked; survived 2M board gf2_fast RIS" } } }, "provenance": { "authors": [ "@MicheldeMessieres" ], "origin": "submission", "novelty": "unknown", "model": "Claude Opus 5.5 (Claude Code)", "construction": "Pair-partition CPM CSS code (Okada-Kasai arXiv:2607.14091), (J,L)=(3,8), even cyclic lift P=122 (Z_122), n=8P=976, k=2P+6=250. H[i*P + r, j*P + ((r - E[i][j]) mod P)] = 1 with 3x8 exponent arrays over Z_122: E_x = [[0, 73, 60, 121, 0, 121, 60, 73], [0, 10, 10, 94, 94, 104, 104, 0], [0, 53, 0, 103, 46, 53, 46, 103]], E_z = [[0, 0, 0, 0, 0, 0, 0, 0], [0, 113, 60, 29, 94, 39, 32, 103], [0, 23, 10, 45, 46, 117, 56, 73]]. The arrays solve the joint design system E_x[i][j] - E_x[i][j'] - E_z[i'][j] + E_z[i'][j'] = 0 (mod 122) for each column pair of the matching M[(i-i') mod 3], M0=(0,4)(1,7)(2,6)(3,5), M1=(0,7)(1,2)(3,4)(5,6), M2=(0,2)(1,5)(3,7)(4,6) (the matchings of the board's [[904,230,22]] by @vprusso); girth 8 (no 4- or 6-cycles in either exponent array). On the even ring Z_122 the rank of H_X, H_Z depends on the exponents mod 2: the free exponents of the design system were restricted to one parity class in which rank H_X = rank H_Z = 363, so k = 2P+6 instead of the generic 2P+4, and the solution was annealed inside that class to reduce the number of 8-cycles. Same construction and design system as the board's [[904,230,22]], [[776,198,20]], [[712,182,20]], [[664,170,18]], [[632,162,18]] (@vprusso), [[808,206,20]] (@pandey-tushar), [[824,210,20]], [[792,202,20]], [[840,214,20]], [[856,218,20]], [[872,222,18]], [[888,226,18]] (@MathysRennela), [[568,146,18]] (@nihoniumqc) and the published [[584,150,18]] (Okada-Kasai).", "references": [ "arXiv:2607.14091", "arXiv:2610.06749", "arXiv:2111.03654", "arXiv:quant-ph/0701020", "https://github.com/kasaikenta/pair-partition-cpm-css-codes" ], "date": "2026-10-14", "notes": "Our contribution: the observation, in our own draws, that on the even ring Z_122 the number of logical qubits depends on the exponents mod 2; the choice of one parity class of the free design-system exponents in which k = 2P + 6 = 250 (two more than the generic 2P + 4); this draw from the 8-cycle-annealed solutions in that class; and the distance screening described in notes/976-250-22.md. The rank of a quasi-cyclic check matrix is set by its common factors with x^P - 1, which has repeated factors over GF(2) when P is even (x^122 - 1 = (x^61 - 1)^2), so extra logical qubits at even lifts are not claimed as a new effect; we found no (3,8) pair-partition CPM entry with k > 2P + 4 in the board's codes, notes and fieldnotes, and did not search the wider literature. Construction and design system: Okada and Kasai, arXiv:2607.14091 (quasi-cyclic CSS LDPC codes from circulant permutation matrices go back to Hagiwara and Imai, arXiv:quant-ph/0701020; @vprusso's entries also cite the lifted-product construction of Panteleev and Kalachev, arXiv:2111.03654). The matchings M0, M1, M2 are the same as in the cell's leader codes/904-230-22.json by @vprusso (found with Claude Fable 5.1); the (3,8) pattern at large lifts and the polishing of the 8-cycle count follow codes/904-230-22.json and codes/712-182-20.json by @vprusso; lifts beyond the paper's P = 73 start with codes/632-162-18.json by @vprusso; the girth-8 hard filter follows codes/808-206-20.json by @pandey-tushar; the score formula kd^2/n = d^2 (1/4 + 1/(2P)) at k = 2P + 4 and the lift cap are from fieldnotes/2026-09-20-screening-traps-at-n-900.md by @e-eight. Other (3,8) weight-8 entries of this family: codes/776-198-20.json and codes/664-170-18.json by @vprusso; codes/824-210-20.json, codes/792-202-20.json, codes/840-214-20.json, codes/856-218-20.json, codes/872-222-18.json and codes/888-226-18.json by @MathysRennela; codes/568-146-18.json by @nihoniumqc; and the published codes/584-150-18.json (Okada-Kasai). Even lifts are already on the board: P = 32 and P = 72 in codes/256-68-12.json and codes/576-148-18.json by @MathysRennela (arrays printed in Maskara et al., arXiv:2610.06749), both at k = 2P + 4. Exact search tool: cpm_distance by Okada and Kasai (github.com/kasaikenta/pair-partition-cpm-css-codes, MIT licence), run from a copy with two small patches of ours: the quasi-cyclic shortcut of one search root per base column, which the original enables only for prime P, enabled for composite P (any lifted word can be cyclically shifted so that its smallest index lies in the copy g = 0 of its base column), and a split option that divides the top-level branches of one root into disjoint pieces. Every logical of this family has even weight (noted in codes/904-230-22.json). Distance is a witness-backed upper bound (confidence upper_bound): the patched cpm_distance, run to completion in 29 pieces per side, found no non-trivial logical of weight 20 or less on either side (3005 CPU-s X, 4698 CPU-s Z), so with even weights d = 22 if the tool and our patch are correct; this was not checked by the board's certifier, and the piece logs are available on request. Heuristic searches found nothing lighter than 22: 2000000 board gf2_fast RIS trials per side (pair depth 8) and Stern information-set decoding, 6 runs of 590 s per side on fresh seeds. Literature novelty is unverified.", "search_budget": { "cpu_hours": 8.0, "ris_trials_per_side": 2000000 } } }