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[[923,174,2]] d ≤
n
923
k
174
d
2
kd²/n
0.754063
w
4
X/Z
1.5
g
0.121
r
2.2361
layers
1
swaps
1

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Distance

X/Z asymmetry 1.5 · d_X ≤ 2, d_Z ≤ 3 · w_X = 4, w_Z = 4 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 2 · witness weight 2 (claimed upper_bound)
witness found by @mathysrennela · verify/heuristic_distance.py:estimate · found at 8000 trials · 2026-10-21
witness operator (support, 2 qubits)
[819, 852]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness found by @mathysrennela · verify/heuristic_distance.py:estimate · found at 8000 trials · survived 8000 trials · 2026-10-21
witness operator (support, 3 qubits)
[306, 307, 338]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking
girth H_X 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 2–4 (mean 3.856) · H_Z 2–4 (mean 3.859)
qubit degrees H_X 1–3 (mean 1.571) · H_Z 1–3 (mean 1.568)
trapping sets H_X (1,1)×397 (2,0)×23 (3,0)×571 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 397 (1,2): 525 (1,3): 1 (2,0): 23 (2,1): 1095 (2,2): 997 (2,3): 2 (3,0): 571 (3,1): 2071 (3,2): 1952 (3,3): 1022 (3,4): 324 (3,5): 1
trapping sets H_Z (1,1)×400 (2,0)×28 (3,0)×574 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 400 (1,2): 522 (1,3): 1 (2,0): 28 (2,1): 1095 (2,2): 989 (2,3): 2 (3,0): 574 (3,1): 2069 (3,2): 1938 (3,3): 1016 (3,4): 321 (3,5): 1
witness diameter X 1.0 · Z 2.2361 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
r = 2.236
X checkZ checkqubit site (923)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release
routing cost 1 nearest-neighbor SWAPs per round in total, at most 1 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

Construction & provenance

provenance submitted through the challenge
novelty novelty not audited
construction C-DELETE-CHECK b344 from 923-173-3.json
model Beam 1.0, MiMo-V2.6-Flash (claimed, not verified)
date 2026-10-22
notes Generated by the priced construction calculus. Distance is reported only with witness-backed upper bounds; lower interval bounds are kept in the separate local ledger. The parent's claimed distance is not inherited wholesale: a randomized (RIS) search over the child finds a lighter logical where one exists, and this entry is filed at the child's own measured upper bound.
family topological (a tag, not a ranking)
locality 2D-local single (computed from the layout)
weight class weight ≤ 4 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[923,174,2]] — one Z-check deletion from 923-173-3.json (C-DELETE-CHECK b344)

Direction & hypothesis

A high-k board region at 923 <= n reachable by deleting one check from 923-173-3.json at little distance cost. Deletion raises k by one at fixed n and fixed qubit set, so any landing point where the frontier has headroom on k is reachable in one move. The aim was the largest d the deletion could keep while still clearing the cell's floor -- here 2, against a parent claim of 3.

What was searched

All 163 CSS board entries with n > 700, not a sample of them: 88,592 essential single check deletions (rows whose removal changes the stabilizer group; dependent rows are dropped, since removing one yields an identical code). 28,913 of those -- 33% -- free a bare qubit and are d=1 by construction; a structural filter on column weights discarded them before any witness search, leaving 59,679 clean deletions across 109 distinct (n, k+1, d, w) tuples.

Witnesses were then re-measured per candidate rather than inherited: 8000 RIS trials per side, seed 0, via the challenge's own heuristic_distance search. At most 3 clean rows were sampled per tuple, so this is a floor on what the pool contains, not an exhaustive sweep.

Evidence trail

  • Parent 923-173-3.json claims d = 3.
  • Unpriced C-DELETE-CHECK carry from the parent (no child witnesses supplied):
  • interval [0, 3]. Contains the measured child distance.

  • Priced transport, d' = min(d, w_del, w_exp): [0, 3] with
  • w_del = 4, w_exp = 5 (RIS upper bounds, 20,000 trials, seed 0; the true minima are NP-hard, Theorem 24.1, so both are over-estimates).

  • Seed-0 estimate over the child: 2 per side, 8000 trials/side.
  • Gate at seed 5: passed, not refuted, no duplicate, board-advancing on no axis in particular.
  • Advancement: nothing dominates it in its cell.

Dead ends

  • Distance does not survive deletion. The parent's claim was never carried forward: on 14 of
  • the 18 filed entries the child measured strictly below the parent, so inheriting the parent's number would have been an over-claim that the refutation gate catches.

  • 28,913 of 88,592 essential deletions free a bare qubit and land at d=1; only 4 of the 18
  • filed entries are of that kind, the rest being shadowed by d=2 siblings from the same parent.

  • 22 candidates in the working set passed the gate but were dominated by a batch-mate at the
  • same (n, k, w), so 40 were reduced to 18 before filing.

  • Not attempted at this scale: C-ADD-CHECK strict improvement and E-GRAFT. The 2026-10-19
  • fieldnote reports 0 strict improvements out of 736 valid additions for n <= 10 and no graft survivors for n <= 8, so neither was worth a sweep at n = 700-1000 without a new idea.

  • C-DIRECT-SUM is unusable against the connectivity gate regardless of scale.

Tools

Models: Beam 1.0, MiMo-V2.6-Flash (matches provenance.model).

  • Structural work: the challenge's own gf2 and verify/heuristic_distance.py, plus an
  • independent nullspace test for row essentiality.

  • Verification: verify/qldpc_verify.py and verify/check_codes_format.py, unchanged, both
  • passing locally on the committed file.

  • Interval transport: the priced construction calculus at github.com/MathysRennela/code-calculus-dev @ 661d1e0, machine/priced_calculus.py. Its CodeState.delete_check
  • reproduces this entry's child state exactly from the parent, and its Independent(...) premise agreed with the independent nullspace check on all 18 pairs.

The calculus run is evidence against OBL-PRICED-INTERVAL-SOUNDNESS, not confirmation of it: containment held on 18/18 for both the unpriced carry and the priced interval, but the prices themselves are over-estimates and no lower endpoint was exercised (the ledger's lower bound stays 0 throughout).

Reproduction

Take 923-173-3.json from codes/, remove the Z-check whose support is [851, 852, 883, 884], and set k = 174. The result is codes/923-174-2.json. Then:

python verify/qldpc_verify.py codes/923-174-2.json
python verify/check_codes_format.py --files codes/923-174-2.json

Both exit 0. The distance witnesses in codes/923-174-2.json are the acceptance argument; they are upper bounds only.

Parity checks

X-checks 376 (max weight 4) · Z-checks 375 (max weight 4)
H_X (376 checks, sparse supports)
[18, 19] [0, 1, 20, 21] [2, 3, 22, 23] [27, 28] [5, 6, 29, 30] [7, 8, 31, 32] [36, 37] [10, 11, 38, 39] [12, 13, 40, 41] [16, 17, 46, 47] [19, 20, 50, 51] [21, 22, 52, 53] [23, 24, 54, 55] [25, 26, 56, 57] [28, 29, 60, 61] [30, 31, 62, 63] [32, 33, 64, 65] [34, 35, 66, 67] [37, 38, 70, 71] [39, 40, 72, 73] [41, 42, 74, 75] [43, 44, 76, 77] [45, 46, 78, 79] [47, 48, 80, 81] [49, 50, 82, 83] [53, 54, 86, 87] [55, 56, 88, 89] [57, 58, 90, 91] [59, 60, 92, 93] [63, 64, 96, 97] [65, 66, 98, 99] [67, 68, 100, 101] [69, 70, 102, 103] [73, 74, 106, 107] [75, 76, 108, 109] [77, 110, 111] [79, 80, 113, 114] [81, 82, 115, 116] [83, 84, 117, 118] [85, 86, 119, 120] [89, 90, 123, 124] [91, 92, 125, 126] [93, 94, 127, 128] [95, 96, 129, 130] [99, 100, 133, 134] [101, 102, 135, 136] [103, 104, 137, 138] [105, 106, 139, 140] [109, 110, 143, 144] [112, 113, 145, 146] [116, 117, 149, 150] [118, 119, 151, 152] [120, 121, 153, 154] [122, 123, 155, 156] [126, 127, 159, 160] [128, 129, 161, 162] [130, 131, 163, 164] [132, 133, 165, 166] [136, 137, 169, 170] [138, 139, 171, 172] [140, 141, 173, 174] [142, 143, 175, 176] [146, 147, 179, 180] [148, 149, 181, 182] [152, 153, 185, 186] [154, 155, 187, 188] [156, 157, 189, 190] [158, 159, 191, 192] [162, 163, 195, 196] [164, 165, 197, 198] [166, 167, 199, 200] [168, 169, 201, 202] [172, 173, 205, 206] [174, 175, 207, 208] [176, 177, 209, 210] [179, 212] [180, 181, 213, 214] [182, 183, 215, 216] [184, 185, 217, 218] [188, 189, 221, 222] [190, 191, 223, 224] [192, 193, 225, 226] [194, 195, 227, 228] [198, 199, 231, 232] [200, 201, 233, 234] [202, 203, 235, 236] [204, 205, 237, 238] [208, 209, 241, 242] [214, 215, 246, 247] [216, 217, 248, 249] [218, 219, 250, 251] [220, 221, 252, 253] [224, 225, 256, 257] [226, 227, 258, 259] [228, 229, 260, 261] [230, 231, 262, 263] [234, 235, 266, 267] [236, 237, 268, 269] [238, 239, 270, 271] [240, 241, 272, 273] [245, 246, 276, 277] [249, 250, 280, 281] [251, 252, 282, 283] [253, 254, 284, 285] [255, 256, 286, 287] [259, 260, 290, 291] [261, 262, 292, 293] [263, 264, 294, 295] [265, 266, 296, 297] [269, 270, 300, 301] [271, 272, 302, 303] [273, 274, 304, 305] [275, 276, 308, 309] [277, 278, 310, 311] [279, 280, 312, 313] [283, 284, 316, 317] [285, 286, 318, 319] [287, 288, 320, 321] [289, 290, 322, 323] [293, 294, 326, 327] [295, 296, 328, 329] [297, 298, 330, 331] [299, 300, 332, 333] [303, 304, 336, 337] [305, 306, 338, 339] [309, 310, 343, 344] [311, 312, 345, 346] [313, 314, 347, 348] [315, 316, 349, 350] [319, 320, 353, 354] [321, 322, 355, 356] [323, 324, 357, 358] [325, 326, 359, 360] [329, 330, 363, 364] [331, 332, 365, 366] [333, 334, 367, 368] [335, 336, 369, 370] [339, 340, 373, 374] [342, 343, 376, 377] [346, 347, 380, 381] [348, 349, 382, 383] [350, 351, 384, 385] [352, 353, 386, 387] [356, 357, 390, 391] [358, 359, 392, 393] [360, 361, 394, 395] [362, 363, 396, 397] [366, 367, 400, 401] [368, 369, 402, 403] [370, 371, 404, 405] [372, 373, 406, 407] [375, 376, 408, 409] [377, 378, 410, 411] [379, 380, 412, 413] [383, 384, 416, 417] [385, 386, 418, 419] [387, 388, 420, 421] [389, 390, 422, 423] [393, 394, 426, 427] [395, 396, 428, 429] [397, 398, 430, 431] [399, 400, 432, 433] [403, 404, 436, 437] [405, 406, 438, 439] [407, 440, 441] [409, 410, 443, 444] [411, 412, 445, 446] [413, 414, 447, 448] [415, 416, 449, 450] [419, 420, 453, 454] [421, 422, 455, 456] [423, 424, 457, 458] [425, 426, 459, 460] [429, 430, 463, 464] [431, 432, 465, 466] [433, 434, 467, 468] [435, 436, 469, 470] [439, 440, 473, 474] [442, 443, 475, 476] [446, 447, 479, 480] [448, 449, 481, 482] [450, 451, 483, 484] [452, 453, 485, 486] [456, 457, 489, 490] [458, 459, 491, 492] [460, 461, 493, 494] [462, 463, 495, 496] [466, 467, 499, 500] [468, 469, 501, 502] [470, 471, 503, 504] [472, 473, 505, 506] [476, 477, 509, 510] [478, 479, 511, 512] [482, 483, 515, 516] [484, 485, 517, 518] [486, 487, 519, 520] [488, 489, 521, 522] [492, 493, 525, 526] [494, 495, 527, 528] [496, 497, 529, 530] [498, 499, 531, 532] [502, 503, 535, 536] [504, 505, 537, 538] [506, 507, 539, 540] [509, 542] [510, 511, 543, 544] [512, 513, 545, 546] [514, 515, 547, 548] [518, 519, 551, 552] [520, 521, 553, 554] [522, 523, 555, 556] [524, 525, 557, 558] [528, 529, 561, 562] [530, 531, 563, 564] [532, 533, 565, 566] [534, 535, 567, 568] [538, 539, 571, 572] [540, 541, 573, 574] [544, 545, 576, 577] [546, 547, 578, 579] [548, 549, 580, 581] [550, 551, 582, 583] [554, 555, 586, 587] [556, 557, 588, 589] [558, 559, 590, 591] [560, 561, 592, 593] [564, 565, 596, 597] [566, 567, 598, 599] [568, 569, 600, 601] [570, 571, 602, 603] [575, 576, 606, 607] [579, 580, 610, 611] [581, 582, 612, 613] [583, 584, 614, 615] [585, 586, 616, 617] [589, 590, 620, 621] [591, 592, 622, 623] [593, 594, 624, 625] [595, 596, 626, 627] [599, 600, 630, 631] [601, 602, 632, 633] [603, 604, 634, 635] [636, 637] [605, 606, 638, 639] [607, 608, 640, 641] [609, 610, 642, 643] [613, 614, 646, 647] [615, 616, 648, 649] [617, 618, 650, 651] [619, 620, 652, 653] [623, 624, 656, 657] [625, 626, 658, 659] [627, 628, 660, 661] [629, 630, 662, 663] [633, 634, 666, 667] [635, 636, 668, 669] [639, 640, 673, 674] [641, 642, 675, 676] [643, 644, 677, 678] [645, 646, 679, 680] [649, 650, 683, 684] [651, 652, 685, 686] [653, 654, 687, 688] [655, 656, 689, 690] [659, 660, 693, 694] [661, 662, 695, 696] [663, 664, 697, 698] [665, 666, 699, 700] [669, 670, 703, 704] [672, 673, 706, 707] [676, 677, 710, 711] [678, 679, 712, 713] [680, 681, 714, 715] [682, 683, 716, 717] [686, 687, 720, 721] [688, 689, 722, 723] [690, 691, 724, 725] [692, 693, 726, 727] [696, 697, 730, 731] [698, 699, 732, 733] [700, 701, 734, 735] [702, 703, 736, 737] [705, 706, 738, 739] [707, 708, 740, 741] [709, 710, 742, 743] [713, 714, 746, 747] [715, 716, 748, 749] [717, 718, 750, 751] [719, 720, 752, 753] [723, 724, 756, 757] [725, 726, 758, 759] [727, 728, 760, 761] [729, 730, 762, 763] [733, 734, 766, 767] [735, 736, 768, 769] [737, 770, 771] [739, 740, 773, 774] [741, 742, 775, 776] [743, 744, 777, 778] [745, 746, 779, 780] [749, 750, 783, 784] [751, 752, 785, 786] [753, 754, 787, 788] [755, 756, 789, 790] [759, 760, 793, 794] [761, 762, 795, 796] [763, 764, 797, 798] [765, 766, 799, 800] [769, 770, 803, 804] [772, 773, 805, 806] [776, 777, 809, 810] [778, 779, 811, 812] [780, 781, 813, 814] [782, 783, 815, 816] [786, 787, 819, 820] [788, 789, 821, 822] [790, 791, 823, 824] [792, 793, 825, 826] [796, 797, 829, 830] [798, 799, 831, 832] [800, 801, 833, 834] [802, 803, 835, 836] [806, 807, 839, 840] [808, 809, 841, 842] [812, 813, 845, 846] [814, 815, 847, 848] [816, 817, 849, 850] [818, 819, 851, 852] [822, 823, 855, 856] [824, 825, 857, 858] [826, 827, 859, 860] [828, 829, 861, 862] [832, 833, 865, 866] [834, 835, 867, 868] [836, 837, 869, 870] [839, 872] [840, 841, 873, 874] [842, 843, 875, 876] [844, 845, 877, 878] [848, 849, 880, 881] [850, 851, 882, 883] [852, 853, 884, 885] [854, 855, 886] [858, 859, 888, 889] [860, 861, 890, 891] [862, 863, 892, 893] [864, 865, 894, 895] [868, 869, 897, 898] [870, 871, 899, 900] [874, 875, 901] [876, 877, 902, 903] [878, 904] [879, 880, 905, 906] [883, 884, 909, 910] [885, 886, 911, 912] [887, 888, 913, 914] [891, 892, 916] [893, 894, 917, 918] [896, 897, 920, 921] [906, 907] [3, 4] [508, 541] [13, 14] [341, 375] [838, 871] [8, 9] [671, 705] [908, 909] [178, 211] [15, 45] [921, 922] [914, 915] [636, 637] [306, 307] [178, 210, 243, 244] [895, 919]
H_Z (375 checks, sparse supports)
[16, 17] [1, 2, 21, 22] [3, 4, 23, 24] [25, 26] [6, 7, 30, 31] [8, 9, 32, 33] [34, 35] [11, 12, 39, 40] [13, 14, 41, 42] [43, 44] [15, 16, 45, 46] [18, 19, 49, 50] [20, 21, 51, 52] [22, 23, 53, 54] [24, 25, 55, 56] [27, 28, 59, 60] [29, 30, 61, 62] [31, 32, 63, 64] [33, 34, 65, 66] [36, 37, 69, 70] [38, 39, 71, 72] [40, 41, 73, 74] [42, 43, 75, 76] [46, 47, 79, 80] [48, 49, 81, 82] [50, 51, 83, 84] [52, 53, 85, 86] [56, 57, 89, 90] [58, 59, 91, 92] [60, 61, 93, 94] [62, 63, 95, 96] [66, 67, 99, 100] [68, 69, 101, 102] [70, 71, 103, 104] [72, 73, 105, 106] [76, 77, 109, 110] [78, 79, 112, 113] [80, 81, 114, 115] [82, 83, 116, 117] [86, 87, 120, 121] [88, 89, 122, 123] [90, 91, 124, 125] [92, 93, 126, 127] [96, 97, 130, 131] [98, 99, 132, 133] [100, 101, 134, 135] [102, 103, 136, 137] [106, 107, 140, 141] [108, 109, 142, 143] [110, 111, 144] [113, 114, 146, 147] [117, 118, 150, 151] [119, 120, 152, 153] [121, 122, 154, 155] [123, 124, 156, 157] [127, 128, 160, 161] [129, 130, 162, 163] [131, 132, 164, 165] [133, 134, 166, 167] [137, 138, 170, 171] [139, 140, 172, 173] [141, 142, 174, 175] [143, 144, 176, 177] [147, 148, 180, 181] [149, 150, 182, 183] [151, 152, 184, 185] [153, 154, 186, 187] [157, 158, 190, 191] [159, 160, 192, 193] [161, 162, 194, 195] [163, 164, 196, 197] [167, 168, 200, 201] [169, 170, 202, 203] [171, 172, 204, 205] [173, 174, 206, 207] [177, 178, 210, 211] [179, 180, 212, 213] [181, 182, 214, 215] [183, 184, 216, 217] [187, 188, 220, 221] [189, 190, 222, 223] [191, 192, 224, 225] [193, 194, 226, 227] [197, 198, 230, 231] [199, 200, 232, 233] [201, 202, 234, 235] [203, 204, 236, 237] [207, 208, 240, 241] [209, 210, 242, 243] [213, 214, 245, 246] [217, 218, 249, 250] [219, 220, 251, 252] [221, 222, 253, 254] [223, 224, 255, 256] [227, 228, 259, 260] [229, 230, 261, 262] [231, 232, 263, 264] [233, 234, 265, 266] [237, 238, 269, 270] [239, 240, 271, 272] [241, 242, 273, 274] [243, 244] [246, 247, 277, 278] [248, 249, 279, 280] [250, 251, 281, 282] [252, 253, 283, 284] [256, 257, 287, 288] [258, 259, 289, 290] [260, 261, 291, 292] [262, 263, 293, 294] [266, 267, 297, 298] [268, 269, 299, 300] [270, 271, 301, 302] [272, 273, 303, 304] [275, 308] [276, 277, 309, 310] [278, 279, 311, 312] [280, 281, 313, 314] [284, 285, 317, 318] [286, 287, 319, 320] [288, 289, 321, 322] [290, 291, 323, 324] [294, 295, 327, 328] [296, 297, 329, 330] [298, 299, 331, 332] [300, 301, 333, 334] [304, 305, 337, 338] [306, 307, 339, 340] [308, 309, 342, 343] [310, 311, 344, 345] [314, 315, 348, 349] [316, 317, 350, 351] [318, 319, 352, 353] [320, 321, 354, 355] [324, 325, 358, 359] [326, 327, 360, 361] [328, 329, 362, 363] [330, 331, 364, 365] [334, 335, 368, 369] [336, 337, 370, 371] [338, 339, 372, 373] [341, 342, 375, 376] [345, 346, 379, 380] [347, 348, 381, 382] [349, 350, 383, 384] [351, 352, 385, 386] [355, 356, 389, 390] [357, 358, 391, 392] [359, 360, 393, 394] [361, 362, 395, 396] [365, 366, 399, 400] [367, 368, 401, 402] [369, 370, 403, 404] [371, 372, 405, 406] [376, 377, 409, 410] [378, 379, 411, 412] [380, 381, 413, 414] [382, 383, 415, 416] [386, 387, 419, 420] [388, 389, 421, 422] [390, 391, 423, 424] [392, 393, 425, 426] [396, 397, 429, 430] [398, 399, 431, 432] [400, 401, 433, 434] [402, 403, 435, 436] [406, 407, 439, 440] [408, 409, 442, 443] [410, 411, 444, 445] [412, 413, 446, 447] [416, 417, 450, 451] [418, 419, 452, 453] [420, 421, 454, 455] [422, 423, 456, 457] [426, 427, 460, 461] [428, 429, 462, 463] [430, 431, 464, 465] [432, 433, 466, 467] [436, 437, 470, 471] [438, 439, 472, 473] [440, 441, 474] [443, 444, 476, 477] [447, 448, 480, 481] [449, 450, 482, 483] [451, 452, 484, 485] [453, 454, 486, 487] [457, 458, 490, 491] [459, 460, 492, 493] [461, 462, 494, 495] [463, 464, 496, 497] [467, 468, 500, 501] [469, 470, 502, 503] [471, 472, 504, 505] [473, 474, 506, 507] [477, 478, 510, 511] [479, 480, 512, 513] [481, 482, 514, 515] [483, 484, 516, 517] [487, 488, 520, 521] [489, 490, 522, 523] [491, 492, 524, 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Code ID 923-174-2 · download JSON · raw on GitHub