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"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=116 (Z_116), n=8P=928, k=2P+6=238. H[i*P + r, j*P + ((r - E[i][j]) mod P)] = 1 with 3x8 exponent arrays over Z_116: E_x = [[0, 59, 56, 55, 0, 55, 56, 59], [0, 14, 14, 20, 20, 34, 34, 0], [0, 12, 0, 54, 24, 12, 24, 54]], E_z = [[0, 0, 0, 0, 0, 0, 0, 0], [0, 68, 56, 50, 20, 64, 76, 54], [0, 17, 14, 79, 24, 37, 38, 59]]. 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 116) 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). As in our [[976,250,22]] at P=122, the free exponents of the design system were fixed to one residue class, here mod 4 (116 = 4 x 29), chosen so that the quotient lift over Z_4 has two more logical qubits than generic; then rank H_X = rank H_Z = 345 and k = 2P+6 instead of the generic 2P+4. The rest of the exponents were annealed inside that class to reduce the number of 8-cycles. Same construction and design system as the board's [[976,250,22]] (@MicheldeMessieres), [[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 extension of the residue-class step of our codes/976-250-22.json from mod 2 to mod 4: on Z_116 (116 = 4 x 29) the free design-system exponents were fixed to a residue class mod 4 chosen so that the quotient lift over Z_4 has two more logical qubits than generic, which carries over to k = 2P + 6 = 238 at P = 116; this draw from the 8-cycle-annealed solutions in that class; and the distance screening described in notes/928-238-22.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^116 - 1 = (x^29 - 1)^4), so extra logical qubits at even lifts are not claimed as a new effect. 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 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/976-250-22.json by @MicheldeMessieres; 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 at k = 2P + 4: codes/256-68-12.json and codes/576-148-18.json by @MathysRennela (arrays printed in Maskara et al., arXiv:2610.06749). Exact search tool: cpm_distance by Okada and Kasai (github.com/kasaikenta/pair-partition-cpm-css-codes, MIT licence), run from the same patched copy described in codes/976-250-22.json (quasi-cyclic root shortcut enabled for composite P; a split option for 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 (3069 CPU-s X, 5223 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 (20 s, then 240 s per side). Literature novelty is unverified.", "search_budget": { "cpu_hours": 53.0, "ris_trials_per_side": 2000000 } } }