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"witness_provenance": { "found_by": [ "@MicheldeMessieres" ], "date": "2026-10-22", "found_at_samples": 155739, "tool": "Stern ISD (own C++), numpy-checked; board heuristic_distance 'inconclusive'", "survived_samples": 400000 } }, "Z": { "value": 29, "confidence": "upper_bound", "witness": [ 5, 50, 84, 149, 157, 211, 220, 281, 335, 337, 424, 453, 501, 526, 570, 600, 602, 608, 681, 740, 755, 793, 833, 861, 894, 928, 949, 956, 973 ], "witness_provenance": { "found_by": [ "@MicheldeMessieres" ], "date": "2026-10-22", "found_at_samples": 156189, "tool": "Stern ISD (own C++), numpy-checked; board heuristic_distance 'inconclusive'", "survived_samples": 400000 } } }, "provenance": { "authors": [ "@MicheldeMessieres" ], "origin": "submission", "novelty": "unknown", "model": "Claude Opus 5.5 (Claude Code)", "construction": "Irregular pair-partition CPM CSS code: the pair-partition commutation rule of Okada-Kasai (arXiv:2607.14091) on an irregular 4x10 'pair-complement' base template instead of the complete JxL one, cyclic lift P=100 (Z_100), n=10P=1000, design rate 1-(4+4)/10=1/5, k=200 (no extra dependencies). Template: the 10 block columns form 5 column pairs {0,1},{2,3},{4,5},{6,7},{8,9}; each of the 4 block rows (per side) contains all pairs but one, so every check has weight 8; the X rows omit pairs {6,7}, {4,5}, {2,3}, {0,1} and the Z rows omit {8,9}, {6,7}, {4,5}, {0,1}, so pair {8,9} is in every X row and pair {2,3} in every Z row (qubit degree 4 there, 3 elsewhere). Block columns per block row: X = [[0, 1, 2, 3, 4, 5, 8, 9], [0, 1, 2, 3, 6, 7, 8, 9], [0, 1, 4, 5, 6, 7, 8, 9], [2, 3, 4, 5, 6, 7, 8, 9]], Z = [[0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 8, 9], [0, 1, 2, 3, 6, 7, 8, 9], [2, 3, 4, 5, 6, 7, 8, 9]]. Every X-row/Z-row overlap (size 6 or 8) is split into column pairs (the pairing below), and the exponents solve x_ia - z_ja - x_ib + z_jb = 0 (mod 100) 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*P + r, c*P + ((r + s_ic) mod P)] = 1 with exponents s (rows = block rows, '-' = empty cell): S_X = [[15, 5, 59, 24, 35, 57, -, -, 72, 38], [53, 5, 99, 25, -, -, 71, 11, 57, 52], [93, 19, -, -, 94, 51, 43, 90, 58, 37], [-, -, 94, 33, 28, 30, 83, 53, 90, 30]], S_Z = [[14, 59, 13, 39, 34, 72, 25, 72, -, -], [97, 7, 88, 27, 37, 39, -, -, 1, 41], [11, 1, 57, 22, -, -, 38, 8, 53, 19], [-, -, 37, 47, 71, 35, 27, 67, 95, 74]]. Pairing (X-row,Z-row): {\"0,0\":[[0,4],[1,2],[3,5]],\"0,1\":[[0,5],[1,4],[2,8],[3,9]],\"0,2\":[[0,1],[2,3],[8,9]],\"0,3\":[[2,5],[3,8],[4,9]],\"1,0\":[[0,7],[1,6],[2,3]],\"1,1\":[[0,8],[1,3],[2,9]],\"1,2\":[[0,2],[1,8],[3,7],[6,9]],\"1,3\":[[2,8],[3,9],[6,7]],\"2,0\":[[0,5],[1,4],[6,7]],\"2,1\":[[0,9],[1,5],[4,8]],\"2,2\":[[0,7],[1,9],[6,8]],\"2,3\":[[4,7],[5,6],[8,9]],\"3,0\":[[2,7],[3,4],[5,6]],\"3,1\":[[2,3],[4,5],[8,9]],\"3,2\":[[2,8],[3,9],[6,7]],\"3,3\":[[2,4],[3,7],[5,8],[6,9]]}. Tanner girth 6 (X) and 6 (Z).", "references": [ "arXiv:2607.14091", "arXiv:2609.30069", "arXiv:quant-ph/0701020", "arXiv:0801.1361", "arXiv:2111.03654", "arXiv:2609.39395", "https://github.com/kasaikenta/pair-partition-cpm-css-codes" ], "date": "2026-10-23", "notes": "Our contribution: the irregular 4x10 \"pair-complement\" base template on which the pair-partition rule is applied (the 10 block columns form 5 column pairs; each of the 4 block rows per side contains all pairs but one, so every check has weight 8; the X rows and the Z rows leave out different pairs, so a different pair has qubit degree 4 on each side; design rate 1/5); the weight-8 template census that selected it (one side's templates with rows of weight <= 8 and column degrees 3..4, X/Z pairs with even row overlaps, ranked by a permanent-vector upper bound, our generalisation of App. A of arXiv:2609.30069 to sparse templates); the couple solver that found the exponent spaces (an exact depth-first search over the pairings of every X-row/Z-row overlap that prunes a partial pairing as soon as a 4-cycle is forced, run as a large-neighbourhood search from known solutions: about 1,380 pairings, 573 distinct solution spaces up to the template's 192 automorphisms); the choice of this pairing, the lift P = 100 and this exponent draw; and the distance screening and lower-bound computation described in notes/1000-200-29.md. Construction framework, not ours: the pair-partition construction and its paired-difference equations on CPM exponents are Okada and Kasai's (arXiv:2607.14091), where they are applied to complete J x L templates; the design-rate count R = 1 - J_tot/L_stab and the permanent-vector bound are from Lee, Okada, Maskara, Kasai and Zhou (arXiv:2609.30069), whose outlook names quantum-compatible irregular templates as a natural direction; this entry is one instance in that direction. 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 on the complete (3,8) template include codes/912-234-22.json by @jacksongquera, codes/904-230-22.json by @vprusso (which also cites the lifted-product construction of Panteleev and Kalachev, arXiv:2111.03654) and the published codes/584-150-18.json (Okada-Kasai); our codes/990-198-17.json applies the rule to a different irregular template with weight-7 checks. In our check for codes/990-198-17.json (2026-10-22: arXiv:2607.14091, arXiv:2609.30069, the Okada-Kasai project page, web searches) we did not find a pair-partition code on an irregular template; literature novelty is not claimed. Distance: a witness-backed upper bound (confidence upper_bound), d <= 29, with a proved lower bound d >= 25. Upper side: Stern information-set decoding found weight-29 logicals on both sides and nothing lighter in 12 runs per side (30 s to 1 h each; the deepest: 2 seeds x 1 h per side, about 464000 iterations each, 29 on all four). The board's verify/heuristic_distance.py (RIS, orbit pass over 8 symmetry partitions, 400000 gf2_fast trials, seed 3) returned 'inconclusive': its lightest logicals had weight 36 (X) and 34 (Z), so at this size it did not get down to the claimed 29 and found nothing lighter. Lower side: an exhaustive branch-and-bound search (our gexact, built on the branch-and-bound of Okada and Kasai's cpm_distance, github.com/kasaikenta/pair-partition-cpm-css-codes, with one root per orbit of the GF(2)-verified Z_100 shift, the orbit anchoring of Rowshan and Devitt, arXiv:2609.39395, and an extra set-cover bound) found no logical of weight <= 24 on either side (all 10 roots per side complete), so d >= 25; the gap 25..28 is open; details in notes/1000-200-29.md.", "search_budget": { "cpu_hours": 31.7, "candidates_screened": 512, "tool": "own Stern ISD (C++) + board heuristic_distance + gexact (lower bound)" } } }