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[[420,14,22]] d ≤
n
420
k
14
d
22
kd²/n
16.133
w
6
X/Z
1.09

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Distance

X/Z asymmetry 1.09 · d_X ≤ 22, d_Z ≤ 24 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 22 · witness weight 22 (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, 22 qubits)
[37, 50, 56, 91, 117, 145, 158, 171, 184, 199, 252, 255, 261, 321, 324, 325, 329, 390, 393, 397, 398, 403]
d_Z 24 · witness weight 24 (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, 24 qubits)
[3, 4, 13, 55, 59, 72, 82, 86, 128, 129, 131, 132, 150, 197, 198, 205, 274, 283, 335, 339, 348, 352, 410, 411]
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 6 · H_Z 6
qubit degrees H_X 3 · H_Z 3
trapping sets H_X (1,3)×420 (2,4)×3150 (3,3)×420 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 420 (2,4): 3150 (3,3): 420 (3,5): 30240 (3,7): 4200
trapping sets H_Z (1,3)×420 (2,4)×3150 (3,3)×420 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 420 (2,4): 3150 (3,3): 420 (3,5): 30240 (3,7): 4200

Construction & provenance

authors @FarLab
provenance submitted through the challenge
novelty novelty not audited
construction Designed-divisor generalized bicycle code on Z_210, a=[0, 69, 137], b=[0, 13, 67].
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

[[420,14,22]] — check-weight-6 generalized-bicycle code

Direction & hypothesis

Target cell: weight-6 x unrestricted (with raw max check weight w = 6). Family: generalized-bicycle. By selecting algebraic generator supports (a, b) of total weight w = 6 sharing a nontrivial common polynomial/group-algebra divisor, k = 14 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 [210].
  • 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 [[420,14,22]] (a = [0, 69, 137], b = [0, 13, 67]):

| stage | trials/side | lightest logical found | |---|---|---| | stage 1 | 25,000 | d <= 22 | | stage 2 | 150,000 | d <= 22 | | stage 3 | 600,000 | d <= 22 |

Final claim: witness-backed upper bound, d <= 22 (confidence: upper_bound, both sides). Operational efficiency kd^2/n = 16.13. 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(*[210])
HX, HZ = build_2bga(mul, [0, 69, 137], [0, 13, 67])

Parity checks

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