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[[900,182,8]] d ≤
n
900
k
182
d
8
kd²/n
12.942
w
5
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 8, d_Z ≤ 8 · w_X = 5, w_Z = 5 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[374, 425, 566, 626, 645, 756, 797, 871]
d_Z 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[111, 160, 179, 302, 303, 329, 388, 515]
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 10 · H_Z 10 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 5 · H_Z 5
qubit degrees H_X 2 · H_Z 2
trapping sets H_X (1,2)×900 (2,2)×3600 (3,2)×14400 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 900 (2,2): 3600 (3,2): 14400 (3,4): 3600
trapping sets H_Z (1,2)×900 (2,2)×3600 (3,2)×14400 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 900 (2,2): 3600 (3,2): 14400 (3,4): 3600

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction Planar hyperbolic {5,5} code, literature reproduction from github.com/QEC-pages/Quantum_LDPC_Codes @ 1c95489383564e4dc2cce517de00d64d6f2c4f56, Hyperbolic_Codes_Planar.zip, parity-check matrices 5_5/5_5_900_X.mtx (HX) and 5_5/5_5_900_Z.mtx (HZ). CSS code from edge/face/vertex incidence of a finite quotient of the regular hyperbolic tiling.
model DeepSeek V4 Flash 0731 (claimed, not verified)
date 2026-09-18
notes Literature reproduction of a published planar hyperbolic code (github.com/QEC-pages/Quantum_LDPC_Codes). Checked against the live board: not equivalent to any existing entry.
family topological (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight ≤ 6 (computed)

How this code was found

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

[[900,182,8]] — planar hyperbolic {5,5} code

Direction & hypothesis

Target: unrestricted weight-6 frontier for hackathon issue #1155. Reproduce a published planar hyperbolic CSS code absent from the local board. The trusted validator reports board advancement relative to the live board (upstream main f6d87255). Efficiency kd²/n = 182·64/900 = 12.94.

What was searched

Scraped GitHub for downloadable parity-check matrix artifacts. From github.com/QEC-pages/Quantum_LDPC_Codes @ 1c95489383564e4dc2cce517de00d64d6f2c4f56, Hyperbolic_Codes_Planar.zip, this code is the {5,5} planar hyperbolic code with 900 qubits. Parity-check matrices are 5_5/5_5_900_X.mtx (HX, 360×900, weight 5) and 5_5/5_5_900_Z.mtx (HZ, 360×900, weight 5). These are incidence matrices of a finite quotient of the regular {5,5} hyperbolic tiling (edge model).

Evidence trail

The trusted candidate gate returned passed: true, structural verification passed (CSS, connectivity, k=182), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 8, not an exact certificate. The published distance (d=8) is consistent with this run.

Dead ends

Attempted to derive a 2D-local layout (interaction radius ≤ 4.0 single-layer, ≤ 7.0 bilayer) via fold_layout annealing and networkx spectral/spring embeddings. None met the caps. Hyperbolic negative curvature prevents a low-radius Euclidean embedding, so this code stays in the unrestricted cell.

Tools

Union Alpha (maker currently anonymous), Zed coding agent; NumPy, networkx, repository submission builder and trusted validator. Public artifacts fetched by immutable commit URL and treated as data. Bounded local CPU run, no paid compute.

Reproduction

Load 5_5/5_5_900_X.mtx (HX) and 5_5/5_5_900_Z.mtx (HZ) from Hyperbolic_Codes_Planar.zip (Matrix Market coordinate format). Verify CSS commutation H_X H_Z^T = 0 over GF(2); k = 900 - rank(HX) - rank(HZ) = 182. Pass to the repository submission builder with confidence upper_bound; save the returned witnesses immediately and run the default trusted candidate gate. Do not replace the returned distance with the paper's number.

Parity checks

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