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314, 363, 378, 449, 627 ], [ 152, 315, 364, 379, 450, 628 ], [ 153, 316, 365, 380, 451, 629 ], [ 154, 317, 366, 381, 452, 630 ], [ 155, 265, 367, 382, 453, 631 ], [ 156, 266, 368, 383, 454, 632 ], [ 157, 267, 369, 384, 455, 633 ], [ 158, 268, 370, 385, 456, 634 ] ] }, "distance": { "d": 15, "X": { "value": 15, "confidence": "exact", "witness": [ 9, 45, 168, 195, 310, 327, 354, 374, 389, 422, 460, 466, 513, 536, 725 ] }, "Z": { "value": 15, "confidence": "exact", "witness": [ 38, 47, 59, 91, 203, 206, 229, 250, 324, 524, 593, 604, 683, 706, 733 ] } }, "provenance": { "authors": [ "@TimoKellerMath" ], "origin": "submission", "novelty": "new_parameters", "construction": "Pair-partition CPM CSS code (the pair-partition rule and girth rule of Okada-Kasai, arXiv:2607.14091) on a sparse 6x14 template of check weight 6, cyclic lift P = 53 (Z_53), n = 14P = 742, k = 2P = 106, design rate 1/7. Template: 6 block rows per type on 14 block columns, every block row of weight 6, every block column of degree 2 or 3 on each side (8 of degree 3, 6 of degree 2; the degree-2 columns of a side form a connected unicyclic graph on its rows). Block columns per block row: X = [[0,1,2,6,10,12], [0,2,3,4,8,13], [1,2,4,5,7,8], [3,4,6,7,9,10], [5,6,7,11,12,13], [8,9,10,11,12,13]], Z = [[0,1,3,5,9,11], [0,4,8,9,12,13], [0,6,10,11,12,13], [1,2,3,7,8,13], [1,2,4,5,10,12], [2,5,6,7,8,11]]. Qubit u*P + s is position s in Z_53 of block column u; X-check (i,r), r in Z_53, acts on the qubits u*P + ((r + e_iu) mod P), u in X_i, Z-check (j,r) on u*P + ((r + f_ju) mod P), u in Z_j; checks in the order i*P + r. Exponents (u:e per block row): X0: 0:0 1:0 2:0 6:0 10:0 12:0; X1: 0:0 2:20 3:0 4:0 8:0 13:0; X2: 1:0 2:38 4:32 5:0 7:0 8:29; X3: 3:6 4:32 6:0 7:15 9:6 10:14; X4: 5:4 6:0 7:20 11:4 12:3 13:0; X5: 8:35 9:1 10:0 11:1 12:38 13:24; Z0: 0:0 1:0 3:0 5:0 9:0 11:0; Z1: 0:0 4:0 8:50 9:27 12:0 13:50; Z2: 0:0 6:0 10:14 11:15 12:14 13:0; Z3: 1:0 2:0 3:44 7:0 8:44 13:33; Z4: 1:0 2:38 4:18 5:39 10:0 12:38; Z5: 2:0 5:4 6:0 7:15 8:33 11:52. Every overlap of an X-row and a Z-row of the template has 0, 2 or 4 block columns and splits into pairs {u,v} with e_iu - f_ju = e_iv - f_jv, so H_X H_Z^T = 0; the pairing satisfies the girth rule (no pair in two cells of one row or one column of the pairing array), and no two checks of one type share two qubits. rank H_X = rank H_Z = 6P = 318; every check has weight 6; qubit (X, Z)-degrees (2,2) on 106, (2,3) on 212, (3,2) on 212 and (3,3) on 212 qubits. The pairing square complex (template rows and columns, one square per pair) has first Betti number 4 and torsion-free H_1; the exponents are a class of H^1 with values in Z_53.", "references": [ "arXiv:2607.14091", "https://kasai.ict.eng.isct.ac.jp/pair_partition_cpm_css_codes_20260714.html", "arXiv:2609.30069", "arXiv:2609.35601", "arXiv:2610.19451", "arXiv:quant-ph/0701020", "arXiv:0901.4129", "arXiv:1611.07164", "arXiv:2611.08247" ], "date": "2026-11-11", "model": "Claude Opus 5.5", "notes": "Exact distance: d_X = d_Z = 15. Two independently written exhaustive searches (exactA, exactB), anchored at the translations of Z_53, find no nontrivial logical of weight <= 14 on either side (records and the certificate certs/742-106-15.json); dist-m4ri 0.16.1 and tropidist b03ec2d4077f, run without symmetry, give the same value on both sides. Definitions, records and a rebuild script: github.com/TimoKellerMath/quantum-error-correction @ 2a15e549f6f4, pair-partition-and-two-block-codes/742-106-15/ and pair-partition-and-two-block-codes/src/rebuild.py. Novelty claim: new parameters only. No entry of board commit 82a52ee51 has n = 742 and k = 106, no published code with these parameters is known to us (literature search of 2026-11-11: see the note), at that commit no CSS entry dominates or equals it in (n, k, d, w), and it dominates 5 entries (see the note). Construction and attribution: a CPM lift of a sparse 6x14 template (check weight 6, column degrees 2 and 3) whose X/Z row overlaps are made even by pair partitions with the girth rule of Okada and Kasai (arXiv:2607.14091); their codes and Kasai's catalogue use all-one JxL templates with check weight L >= 8, as do arXiv:2609.30069 and arXiv:2609.35601. Relation to the board (export 82a52ee51): CPM lifts whose X/Z row overlaps split into pairs of equal exponent difference are common on the board at check weights 4 to 8, among them two-block and bicycle codes, the [[6N,2N+2,6]] family of arXiv:2610.19451 (all-one 2x6 template, d = 6) and irregular templates such as 260-58-6 and the 14x28 lifts 280-20-8 to 700-50-8 at check weight 6; entries tagged pair-partition-cpm occur only at check weight >= 7 (e.g. 990-198-17, 994-142-40, 972-166-34). In the layouts tested (consecutive or interleaved blocks of the stored qubit order), no entry of check weight 6 is a lift of a template with column degrees 2 and 3 (computed). No entry has these n and k; checked, not equivalent to an existing entry." }, "family": "pair-partition-cpm" }