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numpy-checked; board heuristic_distance 'corroborated'" } }, "Z": { "value": 17, "confidence": "exact", "witness": [ 20, 83, 92, 103, 191, 328, 388, 428, 461, 524, 551, 660, 669, 730, 820, 826, 859 ], "witness_provenance": { "found_by": [ "@MicheldeMessieres" ], "date": "2026-10-22", "found_at_samples": 5621, "survived_samples": 400000, "tool": "Stern ISD (own C++), numpy-checked; board heuristic_distance 'corroborated'" } } }, "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 a sparse, irregular 4x10 base template instead of the complete JxL one, cyclic lift P=99 (Z_99), n=10P=990, design rate 1-(4+4)/10=1/5, k=198 (no extra dependencies). Every check has weight 7; qubit degree 2 on block columns 0,1 and 3 on block columns 2..9, both sides. Template (block columns per block row): X = [[0, 2, 3, 4, 5, 6, 7], [0, 2, 3, 4, 5, 8, 9], [1, 2, 3, 6, 7, 8, 9], [1, 4, 5, 6, 7, 8, 9]], Z = [[0, 2, 3, 4, 5, 6, 8], [0, 2, 3, 4, 5, 7, 9], [1, 2, 5, 6, 7, 8, 9], [1, 3, 4, 6, 7, 8, 9]]; every X-row/Z-row overlap has even size and is split into column pairs (the pairing below), and the exponents solve x_ia - z_ja - x_ib + z_jb = 0 (mod 99) 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 = [[73, -, 58, 92, 93, 57, 56, 90, -, -], [33, -, 32, 14, 54, 69, -, -, 17, 47], [-, 54, 38, 59, -, -, 17, 78, 95, 61], [-, 76, -, -, 46, 40, 86, 10, 48, 6]], S_Z = [[10, -, 47, 68, 69, 46, 92, -, 71, -], [24, -, 70, 5, 14, 8, -, 11, -, 7], [-, 55, 9, -, -, 19, 18, 41, 66, 24], [-, 2, -, 75, 49, -, 12, 73, 12, 9]]. Pairing (X-row,Z-row): {\"0,0\":[[0,6],[2,5],[3,4]],\"0,1\":[[0,5],[2,3],[4,7]],\"0,2\":[[2,7],[5,6]],\"0,3\":[[3,7],[4,6]],\"1,0\":[[0,5],[2,4],[3,8]],\"1,1\":[[0,3],[2,5],[4,9]],\"1,2\":[[2,9],[5,8]],\"1,3\":[[3,9],[4,8]],\"2,0\":[[2,3],[6,8]],\"2,1\":[[2,7],[3,9]],\"2,2\":[[1,6],[2,8],[7,9]],\"2,3\":[[1,9],[3,8],[6,7]],\"3,0\":[[4,8],[5,6]],\"3,1\":[[4,5],[7,9]],\"3,2\":[[1,5],[6,7],[8,9]],\"3,3\":[[1,6],[4,9],[7,8]]}. 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:2306.16400", "arXiv:2609.39395", "https://github.com/kasaikenta/pair-partition-cpm-css-codes" ], "date": "2026-10-22", "notes": "Our contribution: the irregular 4x10 base template A (every check weight 7, qubit degrees 2 and 3, design rate 1/5) on which the pair-partition rule is applied; the template census that selected it (one side's templates enumerated exhaustively for 3 or 4 block rows, row weight <= 7, column degrees 2..3, up to row and column permutation; X/Z pairs with even row overlaps placed by SAT and ranked by a permanent-vector upper bound, our generalisation of App. A of arXiv:2609.30069 to sparse templates); the choice of pairing for each X-row/Z-row overlap; the lift P = 99 and this exponent draw; the distance screening described in notes/990-198-17.md, and the exact-distance computation (the GF(2) check of each automorphism generator and the set-cover bound, added to cpm_distance's branch-and-bound; the restriction to one root per automorphism orbit is the orbit anchoring of Rowshan and Devitt, arXiv:2609.39395). 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 mechanism at weight 8: the (3,8) pair-partition entries codes/904-230-22.json, codes/712-182-20.json and codes/632-162-18.json by @vprusso, codes/824-210-20.json by @MathysRennela, codes/808-206-20.json by @pandey-tushar, codes/568-146-18.json by @nihoniumqc, our codes/928-238-22.json and codes/976-250-22.json, and the published codes/584-150-18.json (Okada-Kasai); @vprusso's entries also cite the lifted-product construction of Panteleev and Kalachev (arXiv:2111.03654); the pairing anneal we used was calibrated on the complete (3,8) template of those entries. The weight-7 entries on the board with the highest kd^2/n are codes/682-20-38.json and codes/762-28-32.json by @FarLab (two-block group-algebra codes, arXiv:2306.16400). In our check (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 or with check weight below 8; literature novelty is not claimed. Distance: confidence exact on both sides; the board shows d <= 17; this code is outside verify/certify.py's current envelope (CONTRIBUTING.md), so the exact value is our evidence, not a certificate. Upper side: Stern information-set decoding found weight-17 logicals on both sides and nothing lighter in 3 seeds x 300 s per side, and the board's verify/heuristic_distance.py (RIS, orbit pass over 5 symmetry partitions, 400000 gf2_fast trials) returned 'corroborated', d 17 on both sides. 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_99 shift, the orbit anchoring of Rowshan and Devitt, arXiv:2609.39395) found no logical of weight <= 16 on either side; cross-checked with no automorphisms and with gexact in cpm_distance's mode (its roots and lower bound); details in notes/990-198-17.md.", "search_budget": { "cpu_hours": 12.2, "candidates_screened": 645, "tool": "own Stern ISD (C++) + board heuristic_distance + gexact (exact, 0.2 CPU-h)" } } }