[[504,10,12]] · QEC Challenge

[[504,10,12]] d ≤

504

kd²/n

2.857

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 12, d_Z ≤ 12 · 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 12 · witness weight 12 (claimed upper_bound)

witness found by @FarLab · research/kit/surrogate.lightest_logical · found at 80 trials · survived 2.4×10⁷ trials · 2026-10-14

witness operator (support, 12 qubits)

[272, 293, 314, 335, 356, 377, 398, 419, 440, 461, 482, 503]

d_Z 12 · witness weight 12 (claimed upper_bound)

witness found by @FarLab · research/kit/surrogate.lightest_logical · found at 80 trials · survived 2.4×10⁷ trials · 2026-10-14

witness operator (support, 12 qubits)

[20, 41, 62, 83, 104, 125, 146, 167, 188, 209, 230, 251]

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 5 · H_Z 5

qubit degrees H_X 2–3 (mean 2.5) · H_Z 2–3 (mean 2.5)

trapping sets H_X (1,2)×252 (2,2)×252 (3,2)×252 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,2): 252 (1,3): 252 (2,2): 252 (2,3): 1512 (2,4): 756 (3,2): 252 (3,3): 4788 (3,4): 8316 (3,5): 3024 (3,6): 1512 (3,7): 252

trapping sets H_Z (1,2)×252 (2,2)×252 (3,2)×252 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,2): 252 (1,3): 252 (2,2): 252 (2,3): 1512 (2,4): 756 (3,2): 252 (3,3): 4788 (3,4): 8316 (3,5): 3024 (3,6): 1512 (3,7): 252

Construction & provenance

authors @FarLab

provenance submitted through the challenge

novelty novelty not audited

construction Designed-divisor 2BGA on Z_252 (w5-e21-M12), a=[0, 21], b=[0, 1, 26], w=5=(2+3).

model Gemini 4 Argon (claimed, not verified)

builds on arXiv:2306.16400

date 2026-10-14

family generalized bicycle (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

[[504,10,12]] — check-weight-5 generalized-bicycle code

Direction & hypothesis

Target cell: weight-6 x unrestricted (with raw max check weight w = 5). Family: generalized-bicycle. By selecting algebraic generator supports (a, b) of total weight w = 5 sharing a nontrivial common polynomial/group-algebra divisor, k = 10 is guaranteed by construction while competing on the 4D Pareto frontier (n, k, d, w) in weight-6 x unrestricted.

What was searched

Candidate generated from systematic designed-divisor / group-algebra enumeration over cyclic group parameters [252].
Screened at 3,000 RIS trials on qldpc-box-1 (verify/gf2_fast C++ backend, 4 threads) and promoted through a rising confirmation ladder up to 600,000 RIS trials/side.

Evidence trail

Confirmation ladder for [[504,10,12]] (a = [0, 21], b = [0, 1, 26]):

| stage | trials/side | lightest logical found | |---|---|---| | stage 1 | 25,000 | d <= 12 | | stage 2 | 150,000 | d <= 12 | | stage 3 | 600,000 | d <= 12 |

Final claim: witness-backed upper bound, d <= 12 (confidence: upper_bound, both sides). Operational efficiency kd^2/n = 2.86. Validated with verify/validate_candidate.py (passed: true, board_advancing: true); literature novelty unverified.

Dead ends

Candidates in the same (n, k, w) bucket whose RIS upper bound dropped below the cell domination floor during the 25k, 150k, or 600k trial rungs were discarded before packaging.

Tools

Model: Gemini 4 Argon. Repo tooling: research/kit/group_algebra.py, research/kit/surrogate.py, research/kit/submit.py, verify/gf2_fast, and verify/validate_candidate.py executed on a Brev g2-standard-4:nvidia-l4:1 instance (qldpc-box-1).

Reproduction

from group_algebra import build_2bga, cyclic_product
mul, _ = cyclic_product(*[252])
HX, HZ = build_2bga(mul, [0, 21], [0, 1, 26])

Parity checks

X-checks 252 (max weight 5) · Z-checks 252 (max weight 5)

H_X (252 checks, sparse supports)

[0, 231, 252, 478, 503] [1, 232, 252, 253, 479] [2, 233, 253, 254, 480] [3, 234, 254, 255, 481] [4, 235, 255, 256, 482] [5, 236, 256, 257, 483] [6, 237, 257, 258, 484] [7, 238, 258, 259, 485] [8, 239, 259, 260, 486] [9, 240, 260, 261, 487] [10, 241, 261, 262, 488] [11, 242, 262, 263, 489] [12, 243, 263, 264, 490] [13, 244, 264, 265, 491] [14, 245, 265, 266, 492] [15, 246, 266, 267, 493] [16, 247, 267, 268, 494] [17, 248, 268, 269, 495] [18, 249, 269, 270, 496] [19, 250, 270, 271, 497] [20, 251, 271, 272, 498] [0, 21, 272, 273, 499] [1, 22, 273, 274, 500] [2, 23, 274, 275, 501] [3, 24, 275, 276, 502] [4, 25, 276, 277, 503] [5, 26, 252, 277, 278] [6, 27, 253, 278, 279] [7, 28, 254, 279, 280] [8, 29, 255, 280, 281] [9, 30, 256, 281, 282] [10, 31, 257, 282, 283] [11, 32, 258, 283, 284] [12, 33, 259, 284, 285] [13, 34, 260, 285, 286] [14, 35, 261, 286, 287] [15, 36, 262, 287, 288] [16, 37, 263, 288, 289] [17, 38, 264, 289, 290] [18, 39, 265, 290, 291] [19, 40, 266, 291, 292] [20, 41, 267, 292, 293] [21, 42, 268, 293, 294] [22, 43, 269, 294, 295] [23, 44, 270, 295, 296] [24, 45, 271, 296, 297] [25, 46, 272, 297, 298] [26, 47, 273, 298, 299] [27, 48, 274, 299, 300] [28, 49, 275, 300, 301] [29, 50, 276, 301, 302] [30, 51, 277, 302, 303] [31, 52, 278, 303, 304] [32, 53, 279, 304, 305] [33, 54, 280, 305, 306] [34, 55, 281, 306, 307] [35, 56, 282, 307, 308] [36, 57, 283, 308, 309] [37, 58, 284, 309, 310] [38, 59, 285, 310, 311] [39, 60, 286, 311, 312] [40, 61, 287, 312, 313] [41, 62, 288, 313, 314] [42, 63, 289, 314, 315] [43, 64, 290, 315, 316] [44, 65, 291, 316, 317] [45, 66, 292, 317, 318] [46, 67, 293, 318, 319] [47, 68, 294, 319, 320] [48, 69, 295, 320, 321] [49, 70, 296, 321, 322] [50, 71, 297, 322, 323] [51, 72, 298, 323, 324] [52, 73, 299, 324, 325] [53, 74, 300, 325, 326] [54, 75, 301, 326, 327] [55, 76, 302, 327, 328] [56, 77, 303, 328, 329] [57, 78, 304, 329, 330] [58, 79, 305, 330, 331] [59, 80, 306, 331, 332] [60, 81, 307, 332, 333] [61, 82, 308, 333, 334] [62, 83, 309, 334, 335] [63, 84, 310, 335, 336] [64, 85, 311, 336, 337] [65, 86, 312, 337, 338] [66, 87, 313, 338, 339] [67, 88, 314, 339, 340] [68, 89, 315, 340, 341] [69, 90, 316, 341, 342] [70, 91, 317, 342, 343] [71, 92, 318, 343, 344] [72, 93, 319, 344, 345] [73, 94, 320, 345, 346] [74, 95, 321, 346, 347] [75, 96, 322, 347, 348] [76, 97, 323, 348, 349] [77, 98, 324, 349, 350] [78, 99, 325, 350, 351] [79, 100, 326, 351, 352] [80, 101, 327, 352, 353] [81, 102, 328, 353, 354] [82, 103, 329, 354, 355] [83, 104, 330, 355, 356] [84, 105, 331, 356, 357] [85, 106, 332, 357, 358] [86, 107, 333, 358, 359] [87, 108, 334, 359, 360] [88, 109, 335, 360, 361] [89, 110, 336, 361, 362] [90, 111, 337, 362, 363] [91, 112, 338, 363, 364] [92, 113, 339, 364, 365] [93, 114, 340, 365, 366] [94, 115, 341, 366, 367] [95, 116, 342, 367, 368] [96, 117, 343, 368, 369] [97, 118, 344, 369, 370] [98, 119, 345, 370, 371] [99, 120, 346, 371, 372] [100, 121, 347, 372, 373] [101, 122, 348, 373, 374] [102, 123, 349, 374, 375] [103, 124, 350, 375, 376] [104, 125, 351, 376, 377] [105, 126, 352, 377, 378] [106, 127, 353, 378, 379] [107, 128, 354, 379, 380] [108, 129, 355, 380, 381] [109, 130, 356, 381, 382] [110, 131, 357, 382, 383] [111, 132, 358, 383, 384] [112, 133, 359, 384, 385] [113, 134, 360, 385, 386] [114, 135, 361, 386, 387] [115, 136, 362, 387, 388] [116, 137, 363, 388, 389] [117, 138, 364, 389, 390] [118, 139, 365, 390, 391] [119, 140, 366, 391, 392] [120, 141, 367, 392, 393] [121, 142, 368, 393, 394] [122, 143, 369, 394, 395] [123, 144, 370, 395, 396] [124, 145, 371, 396, 397] [125, 146, 372, 397, 398] [126, 147, 373, 398, 399] [127, 148, 374, 399, 400] [128, 149, 375, 400, 401] [129, 150, 376, 401, 402] [130, 151, 377, 402, 403] [131, 152, 378, 403, 404] [132, 153, 379, 404, 405] [133, 154, 380, 405, 406] [134, 155, 381, 406, 407] [135, 156, 382, 407, 408] [136, 157, 383, 408, 409] [137, 158, 384, 409, 410] [138, 159, 385, 410, 411] [139, 160, 386, 411, 412] [140, 161, 387, 412, 413] [141, 162, 388, 413, 414] [142, 163, 389, 414, 415] [143, 164, 390, 415, 416] [144, 165, 391, 416, 417] [145, 166, 392, 417, 418] [146, 167, 393, 418, 419] [147, 168, 394, 419, 420] [148, 169, 395, 420, 421] [149, 170, 396, 421, 422] [150, 171, 397, 422, 423] [151, 172, 398, 423, 424] [152, 173, 399, 424, 425] [153, 174, 400, 425, 426] [154, 175, 401, 426, 427] [155, 176, 402, 427, 428] [156, 177, 403, 428, 429] [157, 178, 404, 429, 430] [158, 179, 405, 430, 431] [159, 180, 406, 431, 432] [160, 181, 407, 432, 433] [161, 182, 408, 433, 434] [162, 183, 409, 434, 435] [163, 184, 410, 435, 436] [164, 185, 411, 436, 437] [165, 186, 412, 437, 438] [166, 187, 413, 438, 439] [167, 188, 414, 439, 440] [168, 189, 415, 440, 441] [169, 190, 416, 441, 442] [170, 191, 417, 442, 443] [171, 192, 418, 443, 444] [172, 193, 419, 444, 445] [173, 194, 420, 445, 446] [174, 195, 421, 446, 447] [175, 196, 422, 447, 448] [176, 197, 423, 448, 449] [177, 198, 424, 449, 450] [178, 199, 425, 450, 451] [179, 200, 426, 451, 452] [180, 201, 427, 452, 453] [181, 202, 428, 453, 454] [182, 203, 429, 454, 455] [183, 204, 430, 455, 456] [184, 205, 431, 456, 457] [185, 206, 432, 457, 458] [186, 207, 433, 458, 459] [187, 208, 434, 459, 460] [188, 209, 435, 460, 461] [189, 210, 436, 461, 462] [190, 211, 437, 462, 463] [191, 212, 438, 463, 464] [192, 213, 439, 464, 465] [193, 214, 440, 465, 466] [194, 215, 441, 466, 467] [195, 216, 442, 467, 468] [196, 217, 443, 468, 469] [197, 218, 444, 469, 470] [198, 219, 445, 470, 471] [199, 220, 446, 471, 472] [200, 221, 447, 472, 473] [201, 222, 448, 473, 474] [202, 223, 449, 474, 475] [203, 224, 450, 475, 476] [204, 225, 451, 476, 477] [205, 226, 452, 477, 478] [206, 227, 453, 478, 479] [207, 228, 454, 479, 480] [208, 229, 455, 480, 481] [209, 230, 456, 481, 482] [210, 231, 457, 482, 483] [211, 232, 458, 483, 484] [212, 233, 459, 484, 485] [213, 234, 460, 485, 486] [214, 235, 461, 486, 487] [215, 236, 462, 487, 488] [216, 237, 463, 488, 489] [217, 238, 464, 489, 490] [218, 239, 465, 490, 491] [219, 240, 466, 491, 492] [220, 241, 467, 492, 493] [221, 242, 468, 493, 494] [222, 243, 469, 494, 495] [223, 244, 470, 495, 496] [224, 245, 471, 496, 497] [225, 246, 472, 497, 498] [226, 247, 473, 498, 499] [227, 248, 474, 499, 500] [228, 249, 475, 500, 501] [229, 250, 476, 501, 502] [230, 251, 477, 502, 503]

H_Z (252 checks, sparse supports)

[0, 1, 26, 252, 273] [1, 2, 27, 253, 274] [2, 3, 28, 254, 275] [3, 4, 29, 255, 276] [4, 5, 30, 256, 277] [5, 6, 31, 257, 278] [6, 7, 32, 258, 279] [7, 8, 33, 259, 280] [8, 9, 34, 260, 281] [9, 10, 35, 261, 282] [10, 11, 36, 262, 283] [11, 12, 37, 263, 284] [12, 13, 38, 264, 285] [13, 14, 39, 265, 286] [14, 15, 40, 266, 287] [15, 16, 41, 267, 288] [16, 17, 42, 268, 289] [17, 18, 43, 269, 290] [18, 19, 44, 270, 291] [19, 20, 45, 271, 292] [20, 21, 46, 272, 293] [21, 22, 47, 273, 294] [22, 23, 48, 274, 295] [23, 24, 49, 275, 296] [24, 25, 50, 276, 297] [25, 26, 51, 277, 298] [26, 27, 52, 278, 299] [27, 28, 53, 279, 300] [28, 29, 54, 280, 301] [29, 30, 55, 281, 302] [30, 31, 56, 282, 303] [31, 32, 57, 283, 304] [32, 33, 58, 284, 305] [33, 34, 59, 285, 306] [34, 35, 60, 286, 307] [35, 36, 61, 287, 308] [36, 37, 62, 288, 309] [37, 38, 63, 289, 310] [38, 39, 64, 290, 311] [39, 40, 65, 291, 312] [40, 41, 66, 292, 313] [41, 42, 67, 293, 314] [42, 43, 68, 294, 315] [43, 44, 69, 295, 316] [44, 45, 70, 296, 317] [45, 46, 71, 297, 318] [46, 47, 72, 298, 319] [47, 48, 73, 299, 320] [48, 49, 74, 300, 321] [49, 50, 75, 301, 322] [50, 51, 76, 302, 323] [51, 52, 77, 303, 324] [52, 53, 78, 304, 325] [53, 54, 79, 305, 326] [54, 55, 80, 306, 327] [55, 56, 81, 307, 328] [56, 57, 82, 308, 329] [57, 58, 83, 309, 330] [58, 59, 84, 310, 331] [59, 60, 85, 311, 332] [60, 61, 86, 312, 333] [61, 62, 87, 313, 334] [62, 63, 88, 314, 335] [63, 64, 89, 315, 336] [64, 65, 90, 316, 337] [65, 66, 91, 317, 338] [66, 67, 92, 318, 339] [67, 68, 93, 319, 340] [68, 69, 94, 320, 341] [69, 70, 95, 321, 342] [70, 71, 96, 322, 343] [71, 72, 97, 323, 344] [72, 73, 98, 324, 345] [73, 74, 99, 325, 346] [74, 75, 100, 326, 347] [75, 76, 101, 327, 348] [76, 77, 102, 328, 349] [77, 78, 103, 329, 350] [78, 79, 104, 330, 351] [79, 80, 105, 331, 352] [80, 81, 106, 332, 353] [81, 82, 107, 333, 354] [82, 83, 108, 334, 355] [83, 84, 109, 335, 356] [84, 85, 110, 336, 357] [85, 86, 111, 337, 358] [86, 87, 112, 338, 359] [87, 88, 113, 339, 360] [88, 89, 114, 340, 361] [89, 90, 115, 341, 362] [90, 91, 116, 342, 363] [91, 92, 117, 343, 364] [92, 93, 118, 344, 365] [93, 94, 119, 345, 366] [94, 95, 120, 346, 367] [95, 96, 121, 347, 368] [96, 97, 122, 348, 369] [97, 98, 123, 349, 370] [98, 99, 124, 350, 371] [99, 100, 125, 351, 372] [100, 101, 126, 352, 373] [101, 102, 127, 353, 374] [102, 103, 128, 354, 375] [103, 104, 129, 355, 376] [104, 105, 130, 356, 377] [105, 106, 131, 357, 378] [106, 107, 132, 358, 379] [107, 108, 133, 359, 380] [108, 109, 134, 360, 381] [109, 110, 135, 361, 382] [110, 111, 136, 362, 383] [111, 112, 137, 363, 384] [112, 113, 138, 364, 385] [113, 114, 139, 365, 386] [114, 115, 140, 366, 387] [115, 116, 141, 367, 388] [116, 117, 142, 368, 389] [117, 118, 143, 369, 390] [118, 119, 144, 370, 391] [119, 120, 145, 371, 392] [120, 121, 146, 372, 393] [121, 122, 147, 373, 394] [122, 123, 148, 374, 395] [123, 124, 149, 375, 396] [124, 125, 150, 376, 397] [125, 126, 151, 377, 398] [126, 127, 152, 378, 399] [127, 128, 153, 379, 400] [128, 129, 154, 380, 401] [129, 130, 155, 381, 402] [130, 131, 156, 382, 403] [131, 132, 157, 383, 404] [132, 133, 158, 384, 405] [133, 134, 159, 385, 406] [134, 135, 160, 386, 407] [135, 136, 161, 387, 408] [136, 137, 162, 388, 409] [137, 138, 163, 389, 410] [138, 139, 164, 390, 411] [139, 140, 165, 391, 412] [140, 141, 166, 392, 413] [141, 142, 167, 393, 414] [142, 143, 168, 394, 415] [143, 144, 169, 395, 416] [144, 145, 170, 396, 417] [145, 146, 171, 397, 418] [146, 147, 172, 398, 419] [147, 148, 173, 399, 420] [148, 149, 174, 400, 421] [149, 150, 175, 401, 422] [150, 151, 176, 402, 423] [151, 152, 177, 403, 424] [152, 153, 178, 404, 425] [153, 154, 179, 405, 426] [154, 155, 180, 406, 427] [155, 156, 181, 407, 428] [156, 157, 182, 408, 429] [157, 158, 183, 409, 430] [158, 159, 184, 410, 431] [159, 160, 185, 411, 432] [160, 161, 186, 412, 433] [161, 162, 187, 413, 434] [162, 163, 188, 414, 435] [163, 164, 189, 415, 436] [164, 165, 190, 416, 437] [165, 166, 191, 417, 438] [166, 167, 192, 418, 439] [167, 168, 193, 419, 440] [168, 169, 194, 420, 441] [169, 170, 195, 421, 442] [170, 171, 196, 422, 443] [171, 172, 197, 423, 444] [172, 173, 198, 424, 445] [173, 174, 199, 425, 446] [174, 175, 200, 426, 447] [175, 176, 201, 427, 448] [176, 177, 202, 428, 449] [177, 178, 203, 429, 450] [178, 179, 204, 430, 451] [179, 180, 205, 431, 452] [180, 181, 206, 432, 453] [181, 182, 207, 433, 454] [182, 183, 208, 434, 455] [183, 184, 209, 435, 456] [184, 185, 210, 436, 457] [185, 186, 211, 437, 458] [186, 187, 212, 438, 459] [187, 188, 213, 439, 460] [188, 189, 214, 440, 461] [189, 190, 215, 441, 462] [190, 191, 216, 442, 463] [191, 192, 217, 443, 464] [192, 193, 218, 444, 465] [193, 194, 219, 445, 466] [194, 195, 220, 446, 467] [195, 196, 221, 447, 468] [196, 197, 222, 448, 469] [197, 198, 223, 449, 470] [198, 199, 224, 450, 471] [199, 200, 225, 451, 472] [200, 201, 226, 452, 473] [201, 202, 227, 453, 474] [202, 203, 228, 454, 475] [203, 204, 229, 455, 476] [204, 205, 230, 456, 477] [205, 206, 231, 457, 478] [206, 207, 232, 458, 479] [207, 208, 233, 459, 480] [208, 209, 234, 460, 481] [209, 210, 235, 461, 482] [210, 211, 236, 462, 483] [211, 212, 237, 463, 484] [212, 213, 238, 464, 485] [213, 214, 239, 465, 486] [214, 215, 240, 466, 487] [215, 216, 241, 467, 488] [216, 217, 242, 468, 489] [217, 218, 243, 469, 490] [218, 219, 244, 470, 491] [219, 220, 245, 471, 492] [220, 221, 246, 472, 493] [221, 222, 247, 473, 494] [222, 223, 248, 474, 495] [223, 224, 249, 475, 496] [224, 225, 250, 476, 497] [225, 226, 251, 477, 498] [0, 226, 227, 478, 499] [1, 227, 228, 479, 500] [2, 228, 229, 480, 501] [3, 229, 230, 481, 502] [4, 230, 231, 482, 503] [5, 231, 232, 252, 483] [6, 232, 233, 253, 484] [7, 233, 234, 254, 485] [8, 234, 235, 255, 486] [9, 235, 236, 256, 487] [10, 236, 237, 257, 488] [11, 237, 238, 258, 489] [12, 238, 239, 259, 490] [13, 239, 240, 260, 491] [14, 240, 241, 261, 492] [15, 241, 242, 262, 493] [16, 242, 243, 263, 494] [17, 243, 244, 264, 495] [18, 244, 245, 265, 496] [19, 245, 246, 266, 497] [20, 246, 247, 267, 498] [21, 247, 248, 268, 499] [22, 248, 249, 269, 500] [23, 249, 250, 270, 501] [24, 250, 251, 271, 502] [0, 25, 251, 272, 503]

Code ID 504-10-12 · download JSON · raw on GitHub