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452, 454, 466, 493, 495, 522, 579, 602, 613, 617, 625, 661, 667, 712, 739, 761, 802, 804, 815, 841, 870, 926, 929, 931, 933, 941, 961, 974, 975 ] }, "Z": { "value": 40, "confidence": "upper_bound", "witness": [ 8, 16, 27, 32, 133, 136, 238, 257, 286, 338, 343, 373, 392, 410, 427, 432, 440, 476, 490, 506, 560, 563, 576, 696, 712, 728, 740, 764, 778, 783, 799, 812, 835, 845, 877, 888, 896, 907, 950, 964 ], "witness_provenance": { "found_by": [ "@MicheldeMessieres" ], "date": "2026-10-28", "found_at_samples": 46168, "tool": "isd_fast (Stern, p 2, window 16)", "seeds": [ 1 ] } } }, "provenance": { "authors": [ "@MicheldeMessieres" ], "origin": "submission", "novelty": "unknown", "model": "Claude Opus 5.5 (Claude Code)", "construction": "Pair-partition CPM CSS code: the pair-partition commutation rule of Okada-Kasai (arXiv:2607.14091) on a 6x14 'pair-closed' base template, cyclic lift N=71 (Z_71), n=14N=994, design rate 1-(6+6)/14=1/7, k=142 (no extra dependencies). Template (the same on the X and the Z side): the 14 block columns form 7 column pairs {0,1},{2,3},...,{12,13}; each of the 6 block rows is the union of 4 of the 7 pairs, so every check has weight 8; rows by pairs: [[0, 1, 2, 5], [0, 2, 3, 5], [0, 3, 4, 6], [1, 3, 5, 6], [1, 2, 4, 6], [0, 1, 2, 4]] (pair degrees {0: 4, 1: 4, 2: 4, 3: 3, 4: 3, 5: 3, 6: 3}: qubit degree 3 or 4). Block columns per block row: [[0, 1, 2, 3, 4, 5, 10, 11], [0, 1, 4, 5, 6, 7, 10, 11], [0, 1, 6, 7, 8, 9, 12, 13], [2, 3, 6, 7, 10, 11, 12, 13], [2, 3, 4, 5, 8, 9, 12, 13], [0, 1, 2, 3, 4, 5, 8, 9]]. Every X-row/Z-row overlap is split into column pairs (the pairing below), and the exponents solve x_ia - z_ja - x_ib + z_jb = 0 (mod 71) for every column pair (a,b) in the pairing of X block row i with Z block row j, so H_X H_Z^T = 0 term by term. H[i*N + r, c*N + ((r + s_ic) mod N)] = 1 with exponents s (rows = block rows, '-' = empty cell): S_X = [[62, 40, 7, 40, 37, 52, -, -, -, -, 27, 27, -, -], [41, 0, -, -, 8, 4, 4, 20, -, -, 16, 21, -, -], [49, 32, -, -, -, -, 31, 33, 33, 16, -, -, 6, 65], [-, -, 8, 8, -, -, 0, 49, -, -, 53, 25, 5, 12], [-, -, 29, 15, 70, 0, -, -, 69, 9, -, -, 7, 38], [5, 21, 54, 59, 32, 52, -, -, 68, 13, -, -, -, -]], S_Z = [[9, 63, 25, 63, 47, 67, -, -, -, -, 37, 42, -, -], [49, 13, -, -, 24, 25, 12, 33, -, -, 37, 37, -, -], [17, 66, -, -, -, -, 70, 67, 42, 25, -, -, 23, 11], [-, -, 22, 27, -, -, 68, 46, -, -, 42, 14, 19, 31], [-, -, 32, 13, 10, 6, -, -, 4, 20, -, -, 10, 36], [64, 4, 42, 42, 12, 27, -, -, 48, 59, -, -, -, -]]. Pairing (X-row,Z-row): {\"0,0\":[[0,2],[1,3],[4,10],[5,11]],\"0,1\":[[0,4],[1,5],[10,11]],\"0,2\":[[0,1]],\"0,3\":[[2,10],[3,11]],\"0,4\":[[2,5],[3,4]],\"0,5\":[[0,3],[1,2],[4,5]],\"1,0\":[[0,4],[1,5],[10,11]],\"1,1\":[[0,6],[1,7],[4,11],[5,10]],\"1,2\":[[0,7],[1,6]],\"1,3\":[[6,11],[7,10]],\"1,4\":[[4,5]],\"1,5\":[[0,5],[1,4]],\"2,0\":[[0,1]],\"2,1\":[[0,7],[1,6]],\"2,2\":[[0,6],[1,7],[8,9],[12,13]],\"2,3\":[[6,13],[7,12]],\"2,4\":[[8,13],[9,12]],\"2,5\":[[0,8],[1,9]],\"3,0\":[[2,11],[3,10]],\"3,1\":[[6,11],[7,10]],\"3,2\":[[6,13],[7,12]],\"3,3\":[[2,12],[3,13],[6,7],[10,11]],\"3,4\":[[2,13],[3,12]],\"3,5\":[[2,3]],\"4,0\":[[2,5],[3,4]],\"4,1\":[[4,5]],\"4,2\":[[8,13],[9,12]],\"4,3\":[[2,13],[3,12]],\"4,4\":[[2,12],[3,13],[4,9],[5,8]],\"4,5\":[[2,4],[3,5],[8,9]],\"5,0\":[[0,3],[1,2],[4,5]],\"5,1\":[[0,5],[1,4]],\"5,2\":[[0,9],[1,8]],\"5,3\":[[2,3]],\"5,4\":[[2,4],[3,5],[8,9]],\"5,5\":[[0,2],[1,3],[4,8],[5,9]]}.", "references": [ "arXiv:2607.14091", "arXiv:2609.30069", "arXiv:quant-ph/0701020", "arXiv:0801.1361" ], "date": "2026-10-29", "notes": "Our contribution: the 6x14 pair-closed template (7 column pairs, every block row the union of 4 pairs, design rate 1/7), chosen because a typical-distance estimate of the LDPC ensemble with its column-degree mix (the Gallager weight-distribution estimate; Gallager, Low-Density Parity-Check Codes, MIT Press 1963) put rate 1/7 at the best ratio of predicted distance to the distance a rate-1/7 code needs; the search for its pairings (our couple solver, a search for pairings whose exponent equations keep free dimensions, run under one cyclic symmetry of the template at a time; this pairing is one of the solutions with 3 free exponent dimensions of one template class); the lift N = 71 and this exponent draw; and the distance screening. Construction framework, not ours: the pair-partition construction and its paired-difference equations on CPM exponents are Okada and Kasai's (arXiv:2607.14091); the design-rate count R = 1 - J_tot/L_stab and the permanent-vector upper bound used to rank templates (App. A of arXiv:2609.30069) are from Lee, Okada, Maskara, Kasai and Zhou (arXiv:2609.30069); quasi-cyclic CSS LDPC codes from circulant permutation matrices go back to Hagiwara and Imai (arXiv:quant-ph/0701020), and CSS pairs of two irregular LDPC codes were analysed by Hagiwara and Imai (arXiv:0801.1361). Board precedents of the same commutation rule at weight 8: codes/996-166-36.json by @jacksongquera and our codes/996-166-32.json and codes/996-166-34.json (irregular 5x12 template, rate 1/6), codes/912-234-22.json by @jacksongquera and codes/904-230-22.json by @vprusso (complete (3,8) template). Literature novelty is not claimed. Distance: a witness-backed upper bound (confidence upper_bound), d <= 40 (filed witnesses: X weight 41, Z weight 40; a weight-40 X logical seen in one run was not saved); nothing lighter in about 15.4 independent CPU-hours of Stern-type information-set decoding on both sides (39 runs in all, 21.4 CPU-hours; 15.4 after counting runs that repeat a seed in the second program once; two separate programs, our isd_fast with collision windows 16 and 14 and our isd_trace), lightest per side over all runs X 40, Z 40. No lower bound is claimed, so the true distance and score may be lower. Details in notes/994-142-40.md.", "search_budget": { "candidates_screened": 815, "cpu_hours": 56.0, "wall_clock_hours": 9.0, "tool": "couple solver (csolve.mjs) + lift.mjs + isd_fast / isd_trace funnel", "notes": "Search for this template on 2026-10-28: about 50.5 CPU-h of Stern-type ISD over 109 lifted codes plus about 5 CPU-h of couple search, lifts and 8 s screens of 815 lifts. Not counted: the distance checks of this code after the search (21.4 CPU-h counting every run)." } } }