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[[648,434,3]] d ≤
n
648
k
434
d
3
kd²/n
6.02778
w
12
X/Z
1

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 3, d_Z ≤ 3 · w_X = 12, w_Z = 12 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[280, 282, 302]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[561, 571, 632]
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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 12 · H_Z 12
qubit degrees H_X 2 · H_Z 2
trapping sets H_X (1,2)×648 (2,2)×7128 (3,0)×864 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 648 (2,2): 7128 (3,0): 864 (3,2): 75816 (3,4): 23760
trapping sets H_Z (1,2)×648 (2,2)×7128 (3,0)×864 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 648 (2,2): 7128 (3,0): 864 (3,2): 75816 (3,4): 23760

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction contributed via qldpc submit
model Beam 1.0 (claimed, not verified)
date 2026-11-09
family other (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight > 8 (computed)

How this code was found

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

[[648,434,3]] — FCC K=12 lattice code (L=6)

Direction & hypothesis

Same as L=4: high-rate constant-distance CSS code on the FCC lattice. L=6 gives n=648, k=434, d=3.

What was searched

FCC lattice construction with L=6 (arXiv:2603.20294). Z-stabilizers on vertices, X-stabilizers on octahedral voids, weight 12. d=3 proven by exhaustive elimination of weight<=2 plus weight-3 logicals.

Evidence trail

Distance d=3 (upper bound) with weight-3 witnesses on both sides. Gate passed.

Dead ends

L=2 refuted; L=8 exceeds n cap.

Tools

Reconstructed from arXiv:2603.20294 Appendix A. Model: Beam 1.0.

Reproduction

Build from arXiv:2603.20294 Appendix A with L=6.

Benchmark comparison (optional, Tier B/C)

Not applicable. EOF

Parity checks

X-checks 108 (max weight 12) · Z-checks 108 (max weight 12)
H_X (108 checks, sparse supports)
[4, 6, 8, 10, 17, 19, 21, 23, 37, 39, 156, 158] [16, 18, 20, 22, 29, 31, 33, 35, 47, 49, 164, 166] [5, 7, 9, 11, 28, 30, 32, 34, 57, 59, 172, 174] [0, 2, 8, 9, 41, 43, 45, 60, 62, 64, 67, 69] [12, 14, 20, 21, 40, 42, 44, 51, 53, 55, 77, 79] [24, 26, 32, 33, 50, 52, 54, 61, 63, 65, 87, 89] [36, 38, 44, 45, 70, 72, 74, 81, 83, 85, 97, 99] [46, 48, 54, 55, 80, 82, 84, 91, 93, 95, 107, 109] [56, 58, 64, 65, 71, 73, 75, 90, 92, 94, 117, 119] [66, 68, 74, 75, 101, 103, 105, 120, 122, 124, 127, 129] [76, 78, 84, 85, 100, 102, 104, 111, 113, 115, 137, 139] [86, 88, 94, 95, 110, 112, 114, 121, 123, 125, 147, 149] [96, 98, 104, 105, 130, 132, 134, 141, 143, 145, 157, 159] [106, 108, 114, 115, 140, 142, 144, 151, 153, 155, 165, 167] [116, 118, 124, 125, 131, 133, 135, 150, 152, 154, 173, 175] [1, 3, 10, 11, 126, 128, 134, 135, 161, 163, 176, 178] [13, 15, 22, 23, 136, 138, 144, 145, 160, 162, 169, 171] [25, 27, 34, 35, 146, 148, 154, 155, 168, 170, 177, 179] [0, 1, 4, 5, 183, 185, 187, 198, 200, 202, 205, 276] [12, 13, 16, 17, 182, 184, 186, 191, 193, 195, 211, 280] [24, 25, 28, 29, 190, 192, 194, 199, 201, 203, 217, 284] [36, 37, 40, 41, 180, 184, 185, 206, 208, 213, 215, 223] [46, 47, 50, 51, 188, 192, 193, 212, 214, 219, 221, 229] [56, 57, 60, 61, 196, 200, 201, 207, 209, 218, 220, 235] [66, 67, 70, 71, 204, 208, 209, 225, 227, 236, 238, 241] [76, 77, 80, 81, 210, 214, 215, 224, 226, 231, 233, 247] [86, 87, 90, 91, 216, 220, 221, 230, 232, 237, 239, 253] [96, 97, 100, 101, 222, 226, 227, 242, 244, 249, 251, 259] [106, 107, 110, 111, 228, 232, 233, 248, 250, 255, 257, 265] [116, 117, 120, 121, 234, 238, 239, 243, 245, 254, 256, 271] [126, 127, 130, 131, 240, 244, 245, 261, 263, 272, 274, 277] [136, 137, 140, 141, 246, 250, 251, 260, 262, 267, 269, 281] [146, 147, 150, 151, 252, 256, 257, 266, 268, 273, 275, 285] [156, 157, 160, 161, 181, 186, 187, 258, 262, 263, 278, 283] [164, 165, 168, 169, 189, 194, 195, 264, 268, 269, 282, 287] [172, 173, 176, 177, 197, 202, 203, 270, 274, 275, 279, 286] [180, 181, 182, 183, 290, 292, 294, 299, 301, 303, 313, 384] [188, 189, 190, 191, 298, 300, 302, 307, 309, 311, 319, 388] [196, 197, 198, 199, 291, 293, 295, 306, 308, 310, 325, 392] [204, 205, 206, 207, 288, 292, 293, 315, 317, 326, 328, 331] [210, 211, 212, 213, 296, 300, 301, 314, 316, 321, 323, 337] [216, 217, 218, 219, 304, 308, 309, 320, 322, 327, 329, 343] [222, 223, 224, 225, 312, 316, 317, 332, 334, 339, 341, 349] [228, 229, 230, 231, 318, 322, 323, 338, 340, 345, 347, 355] [234, 235, 236, 237, 324, 328, 329, 333, 335, 344, 346, 361] [240, 241, 242, 243, 330, 334, 335, 351, 353, 362, 364, 367] [246, 247, 248, 249, 336, 340, 341, 350, 352, 357, 359, 373] [252, 253, 254, 255, 342, 346, 347, 356, 358, 363, 365, 379] [258, 259, 260, 261, 348, 352, 353, 368, 370, 375, 377, 385] [264, 265, 266, 267, 354, 358, 359, 374, 376, 381, 383, 389] [270, 271, 272, 273, 360, 364, 365, 369, 371, 380, 382, 393] [276, 277, 278, 279, 289, 294, 295, 366, 370, 371, 387, 394] [280, 281, 282, 283, 297, 302, 303, 372, 376, 377, 386, 391] [284, 285, 286, 287, 305, 310, 311, 378, 382, 383, 390, 395] [288, 289, 290, 291, 399, 401, 403, 414, 416, 418, 421, 492] [296, 297, 298, 299, 398, 400, 402, 407, 409, 411, 427, 496] [304, 305, 306, 307, 406, 408, 410, 415, 417, 419, 433, 500] [312, 313, 314, 315, 396, 400, 401, 422, 424, 429, 431, 439] [318, 319, 320, 321, 404, 408, 409, 428, 430, 435, 437, 445] [324, 325, 326, 327, 412, 416, 417, 423, 425, 434, 436, 451] [330, 331, 332, 333, 420, 424, 425, 441, 443, 452, 454, 457] [336, 337, 338, 339, 426, 430, 431, 440, 442, 447, 449, 463] [342, 343, 344, 345, 432, 436, 437, 446, 448, 453, 455, 469] [348, 349, 350, 351, 438, 442, 443, 458, 460, 465, 467, 475] [354, 355, 356, 357, 444, 448, 449, 464, 466, 471, 473, 481] [360, 361, 362, 363, 450, 454, 455, 459, 461, 470, 472, 487] [366, 367, 368, 369, 456, 460, 461, 477, 479, 488, 490, 493] [372, 373, 374, 375, 462, 466, 467, 476, 478, 483, 485, 497] [378, 379, 380, 381, 468, 472, 473, 482, 484, 489, 491, 501] [384, 385, 386, 387, 397, 402, 403, 474, 478, 479, 494, 499] [388, 389, 390, 391, 405, 410, 411, 480, 484, 485, 498, 503] [392, 393, 394, 395, 413, 418, 419, 486, 490, 491, 495, 502] [396, 397, 398, 399, 506, 508, 510, 515, 517, 519, 529, 600] [404, 405, 406, 407, 514, 516, 518, 523, 525, 527, 535, 604] [412, 413, 414, 415, 507, 509, 511, 522, 524, 526, 541, 608] [420, 421, 422, 423, 504, 508, 509, 531, 533, 542, 544, 547] [426, 427, 428, 429, 512, 516, 517, 530, 532, 537, 539, 553] [432, 433, 434, 435, 520, 524, 525, 536, 538, 543, 545, 559] [438, 439, 440, 441, 528, 532, 533, 548, 550, 555, 557, 565] [444, 445, 446, 447, 534, 538, 539, 554, 556, 561, 563, 571] [450, 451, 452, 453, 540, 544, 545, 549, 551, 560, 562, 577] [456, 457, 458, 459, 546, 550, 551, 567, 569, 578, 580, 583] [462, 463, 464, 465, 552, 556, 557, 566, 568, 573, 575, 589] [468, 469, 470, 471, 558, 562, 563, 572, 574, 579, 581, 595] [474, 475, 476, 477, 564, 568, 569, 584, 586, 591, 593, 601] [480, 481, 482, 483, 570, 574, 575, 590, 592, 597, 599, 605] [486, 487, 488, 489, 576, 580, 581, 585, 587, 596, 598, 609] [492, 493, 494, 495, 505, 510, 511, 582, 586, 587, 603, 610] [496, 497, 498, 499, 513, 518, 519, 588, 592, 593, 602, 607] [500, 501, 502, 503, 521, 526, 527, 594, 598, 599, 606, 611] [2, 3, 6, 7, 504, 505, 506, 507, 613, 615, 620, 622] [14, 15, 18, 19, 512, 513, 514, 515, 612, 614, 617, 619] [26, 27, 30, 31, 520, 521, 522, 523, 616, 618, 621, 623] [38, 39, 42, 43, 528, 529, 530, 531, 612, 613, 624, 627] [48, 49, 52, 53, 534, 535, 536, 537, 616, 617, 626, 629] [58, 59, 62, 63, 540, 541, 542, 543, 620, 621, 625, 628] [68, 69, 72, 73, 546, 547, 548, 549, 624, 625, 631, 634] [78, 79, 82, 83, 552, 553, 554, 555, 626, 627, 630, 633] [88, 89, 92, 93, 558, 559, 560, 561, 628, 629, 632, 635] [98, 99, 102, 103, 564, 565, 566, 567, 630, 631, 636, 639] [108, 109, 112, 113, 570, 571, 572, 573, 632, 633, 638, 641] [118, 119, 122, 123, 576, 577, 578, 579, 634, 635, 637, 640] [128, 129, 132, 133, 582, 583, 584, 585, 636, 637, 643, 646] [138, 139, 142, 143, 588, 589, 590, 591, 638, 639, 642, 645] [148, 149, 152, 153, 594, 595, 596, 597, 640, 641, 644, 647] [158, 159, 162, 163, 600, 601, 602, 603, 614, 615, 642, 643] [166, 167, 170, 171, 604, 605, 606, 607, 618, 619, 644, 645] [174, 175, 178, 179, 608, 609, 610, 611, 622, 623, 646, 647]
H_Z (108 checks, sparse supports)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23] [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35] [8, 21, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45] [20, 33, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55] [9, 32, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65] [45, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75] [44, 55, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85] [54, 65, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95] [74, 85, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105] [84, 95, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115] [75, 94, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125] [105, 124, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135] [104, 115, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145] [114, 125, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155] [10, 23, 134, 145, 156, 157, 158, 159, 160, 161, 162, 163] [22, 35, 144, 155, 164, 165, 166, 167, 168, 169, 170, 171] [11, 34, 135, 154, 172, 173, 174, 175, 176, 177, 178, 179] [4, 17, 37, 156, 180, 181, 182, 183, 184, 185, 186, 187] [16, 29, 47, 164, 188, 189, 190, 191, 192, 193, 194, 195] [5, 28, 57, 172, 196, 197, 198, 199, 200, 201, 202, 203] [0, 41, 60, 67, 185, 200, 204, 205, 206, 207, 208, 209] [12, 40, 51, 77, 184, 193, 210, 211, 212, 213, 214, 215] [24, 50, 61, 87, 192, 201, 216, 217, 218, 219, 220, 221] [36, 70, 81, 97, 208, 215, 222, 223, 224, 225, 226, 227] [46, 80, 91, 107, 214, 221, 228, 229, 230, 231, 232, 233] [56, 71, 90, 117, 209, 220, 234, 235, 236, 237, 238, 239] [66, 101, 120, 127, 227, 238, 240, 241, 242, 243, 244, 245] [76, 100, 111, 137, 226, 233, 246, 247, 248, 249, 250, 251] [86, 110, 121, 147, 232, 239, 252, 253, 254, 255, 256, 257] [96, 130, 141, 157, 244, 251, 258, 259, 260, 261, 262, 263] [106, 140, 151, 165, 250, 257, 264, 265, 266, 267, 268, 269] [116, 131, 150, 173, 245, 256, 270, 271, 272, 273, 274, 275] [1, 126, 161, 176, 187, 202, 263, 274, 276, 277, 278, 279] [13, 136, 160, 169, 186, 195, 262, 269, 280, 281, 282, 283] [25, 146, 168, 177, 194, 203, 268, 275, 284, 285, 286, 287] [183, 198, 205, 276, 288, 289, 290, 291, 292, 293, 294, 295] [182, 191, 211, 280, 296, 297, 298, 299, 300, 301, 302, 303] [190, 199, 217, 284, 304, 305, 306, 307, 308, 309, 310, 311] [180, 206, 213, 223, 292, 301, 312, 313, 314, 315, 316, 317] [188, 212, 219, 229, 300, 309, 318, 319, 320, 321, 322, 323] [196, 207, 218, 235, 293, 308, 324, 325, 326, 327, 328, 329] [204, 225, 236, 241, 317, 328, 330, 331, 332, 333, 334, 335] [210, 224, 231, 247, 316, 323, 336, 337, 338, 339, 340, 341] [216, 230, 237, 253, 322, 329, 342, 343, 344, 345, 346, 347] [222, 242, 249, 259, 334, 341, 348, 349, 350, 351, 352, 353] [228, 248, 255, 265, 340, 347, 354, 355, 356, 357, 358, 359] [234, 243, 254, 271, 335, 346, 360, 361, 362, 363, 364, 365] [240, 261, 272, 277, 353, 364, 366, 367, 368, 369, 370, 371] [246, 260, 267, 281, 352, 359, 372, 373, 374, 375, 376, 377] [252, 266, 273, 285, 358, 365, 378, 379, 380, 381, 382, 383] [181, 258, 278, 283, 294, 303, 370, 377, 384, 385, 386, 387] [189, 264, 282, 287, 302, 311, 376, 383, 388, 389, 390, 391] [197, 270, 279, 286, 295, 310, 371, 382, 392, 393, 394, 395] [290, 299, 313, 384, 396, 397, 398, 399, 400, 401, 402, 403] [298, 307, 319, 388, 404, 405, 406, 407, 408, 409, 410, 411] [291, 306, 325, 392, 412, 413, 414, 415, 416, 417, 418, 419] [288, 315, 326, 331, 401, 416, 420, 421, 422, 423, 424, 425] [296, 314, 321, 337, 400, 409, 426, 427, 428, 429, 430, 431] [304, 320, 327, 343, 408, 417, 432, 433, 434, 435, 436, 437] [312, 332, 339, 349, 424, 431, 438, 439, 440, 441, 442, 443] [318, 338, 345, 355, 430, 437, 444, 445, 446, 447, 448, 449] [324, 333, 344, 361, 425, 436, 450, 451, 452, 453, 454, 455] [330, 351, 362, 367, 443, 454, 456, 457, 458, 459, 460, 461] [336, 350, 357, 373, 442, 449, 462, 463, 464, 465, 466, 467] [342, 356, 363, 379, 448, 455, 468, 469, 470, 471, 472, 473] [348, 368, 375, 385, 460, 467, 474, 475, 476, 477, 478, 479] [354, 374, 381, 389, 466, 473, 480, 481, 482, 483, 484, 485] [360, 369, 380, 393, 461, 472, 486, 487, 488, 489, 490, 491] [289, 366, 387, 394, 403, 418, 479, 490, 492, 493, 494, 495] [297, 372, 386, 391, 402, 411, 478, 485, 496, 497, 498, 499] [305, 378, 390, 395, 410, 419, 484, 491, 500, 501, 502, 503] [399, 414, 421, 492, 504, 505, 506, 507, 508, 509, 510, 511] [398, 407, 427, 496, 512, 513, 514, 515, 516, 517, 518, 519] [406, 415, 433, 500, 520, 521, 522, 523, 524, 525, 526, 527] [396, 422, 429, 439, 508, 517, 528, 529, 530, 531, 532, 533] [404, 428, 435, 445, 516, 525, 534, 535, 536, 537, 538, 539] [412, 423, 434, 451, 509, 524, 540, 541, 542, 543, 544, 545] [420, 441, 452, 457, 533, 544, 546, 547, 548, 549, 550, 551] [426, 440, 447, 463, 532, 539, 552, 553, 554, 555, 556, 557] [432, 446, 453, 469, 538, 545, 558, 559, 560, 561, 562, 563] [438, 458, 465, 475, 550, 557, 564, 565, 566, 567, 568, 569] [444, 464, 471, 481, 556, 563, 570, 571, 572, 573, 574, 575] [450, 459, 470, 487, 551, 562, 576, 577, 578, 579, 580, 581] [456, 477, 488, 493, 569, 580, 582, 583, 584, 585, 586, 587] [462, 476, 483, 497, 568, 575, 588, 589, 590, 591, 592, 593] [468, 482, 489, 501, 574, 581, 594, 595, 596, 597, 598, 599] [397, 474, 494, 499, 510, 519, 586, 593, 600, 601, 602, 603] [405, 480, 498, 503, 518, 527, 592, 599, 604, 605, 606, 607] [413, 486, 495, 502, 511, 526, 587, 598, 608, 609, 610, 611] [6, 19, 39, 158, 506, 515, 529, 600, 612, 613, 614, 615] [18, 31, 49, 166, 514, 523, 535, 604, 616, 617, 618, 619] [7, 30, 59, 174, 507, 522, 541, 608, 620, 621, 622, 623] [2, 43, 62, 69, 504, 531, 542, 547, 613, 620, 624, 625] [14, 42, 53, 79, 512, 530, 537, 553, 612, 617, 626, 627] [26, 52, 63, 89, 520, 536, 543, 559, 616, 621, 628, 629] [38, 72, 83, 99, 528, 548, 555, 565, 624, 627, 630, 631] [48, 82, 93, 109, 534, 554, 561, 571, 626, 629, 632, 633] [58, 73, 92, 119, 540, 549, 560, 577, 625, 628, 634, 635] [68, 103, 122, 129, 546, 567, 578, 583, 631, 634, 636, 637] [78, 102, 113, 139, 552, 566, 573, 589, 630, 633, 638, 639] [88, 112, 123, 149, 558, 572, 579, 595, 632, 635, 640, 641] [98, 132, 143, 159, 564, 584, 591, 601, 636, 639, 642, 643] [108, 142, 153, 167, 570, 590, 597, 605, 638, 641, 644, 645] [118, 133, 152, 175, 576, 585, 596, 609, 637, 640, 646, 647] [3, 128, 163, 178, 505, 582, 603, 610, 615, 622, 643, 646] [15, 138, 162, 171, 513, 588, 602, 607, 614, 619, 642, 645] [27, 148, 170, 179, 521, 594, 606, 611, 618, 623, 644, 647]
Code ID 648-434-3 · download JSON · raw on GitHub