[[420,4,22]] · QEC Challenge

[[420,4,22]] d ≤

420

kd²/n

4.61

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 22, d_Z ≤ 22 · 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 22 · witness weight 22 (claimed upper_bound)

witness operator (support, 22 qubits)

[1, 45, 69, 83, 93, 103, 113, 123, 137, 157, 177, 191, 215, 228, 258, 271, 301, 309, 314, 344, 352, 395]

d_Z 22 · witness weight 22 (claimed upper_bound)

witness found by @vprusso · ris_gpu 300M trials/side, pair depth 8, Perlmutter A100 · found at 3×10⁸ trials · 2026-10-09

witness operator (support, 22 qubits)

[18, 19, 23, 61, 100, 104, 143, 147, 186, 190, 243, 253, 267, 277, 297, 335, 341, 345, 351, 375, 389, 409]

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)×210 (2,2)×210 (3,2)×210 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 210 (1,3): 210 (2,2): 210 (2,3): 1260 (2,4): 630 (3,2): 210 (3,3): 3990 (3,4): 6930 (3,5): 2520 (3,6): 1260 (3,7): 210

trapping sets H_Z (1,2)×210 (2,2)×210 (3,2)×210 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 210 (1,3): 210 (2,2): 210 (2,3): 1260 (2,4): 630 (3,2): 210 (3,3): 3990 (3,4): 6930 (3,5): 2520 (3,6): 1260 (3,7): 210

Construction & provenance

authors @cbjuan and @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction Bivariate bicycle code over F_2[x,y]/(x¹⁵-1, y¹⁴-1) with mixed-monomial weight-five polynomials A = 1 + x³ y, B = 1 + x¹⁰ y⁶ + x¹¹ y², lattice (15,14), taken from Table 3 of arXiv:2610.06623v1. The paper reports this presentation with the interval 5 <= d <= 22 (evidence L4+I+B) and no exact certificate. The distance claimed here is an upper bound from this repository's own witness search, not the paper's interval endpoint. Construction by Juan Cruz-Benito (@cbjuan); the independent re-derivation and the distance witnesses are by @MathysRennela.

model human (claimed, not verified)

date 2026-10-06

notes Known parameters: the (n,k,d) triple and the explicit polynomial presentation are published in arXiv:2610.06623v1, Table 3. This entry is an independent reconstruction and re-measurement, not a new construction. The claimed distance is a witness-backed upper bound; the paper states no exact certificate for this row. The paper prints a second (15,14) presentation with the same parameters (A = 1 + x^4 y^3 + x^8 y^11, B = 1 + x^3 y^3); it shares this entry's WL signature, so it is possibly the same code and is not filed separately. Distance revision 2026-10-09 (Perlmutter refutation campaign, witnesses by @vprusso): Z side 23 -> 22 (GPU randomized information-set search (verify/ris_gpu.py, pair depth 8), 300,000,000 trials per side on an NVIDIA A100, seed 1).

family bivariate 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

[[420,4,22]] — weight-5 bivariate bicycle code from arXiv:2610.06623v1 Table 3

Direction & hypothesis

Target: the unrestricted x weight-6 CSS cell, thin k = 4 corner at large n. Screening arXiv:2610.06623v1 left two rows whose distance the paper reports as an interval rather than an exact value, and the board holds neither.

kd^2/n is 4.61, well below the 30.5 the same cell already holds ([[672,20,32]]). This is a thin-corner record, not a strong one.

What was searched

No fresh search. All 10 CSS presentations in Table 3 of arXiv:2610.06623v1 were rebuilt with research/kit/bb.py:build_bb and their (n, k, w) measured from the matrices; all 10 reproduced their claimed parameters. Replicated components were dropped first (the paper's own connectivity audit finds 409 of 1,142 catalogue entries disconnected, and 73.1% of exact classes to be repeated components), then survivors were Pareto-screened against codes/.

Seven Table 3 rows are dominated by existing entries and two are already on the board. This row and [[432,4,23]] are the only undominated survivors.

Evidence trail

The presentation: l = 15, m = 14, A = 1 + x^3 y, `B = 1 + x^10 y^6 + x^11 y^2, so n = 2*15*14 = 420`.

Rebuilt and measured: n = 420, k = 4, max check weight 5, one connected

component.

Distance: the paper reports 5 <= d <= 22 (its L4+I+B evidence) and no

exact certificate. This repository's searches (research/kit/surrogate.py:distance_rand_witness, 4 seeds x 6000 RIS trials, pair_depth=16, gf2_fast backend) returned weight-22 logicals and nothing lighter on the X side.

Submitted as a witness-backed upper bound, d <= 22. It is not

certified exact and is not described as such.

The trusted gate (verify/validate_candidate.py) returns passed: true,

dedup clean, refutation unrefuted over 8000 RIS trials, and novelty.board_advancing: true on advances_by: ["d", "n"].

The second (15,14) presentation

The paper prints two (15,14) presentations with these same parameters and the same interval evidence:

1. A = 1 + x^3 y, B = 1 + x^10 y^6 + x^11 y^2 — filed here. 2. A = 1 + x^4 y^3 + x^8 y^11, B = 1 + x^3 y^3.

They have different fingerprints but the same WL signature, which is the board's own "possibly equivalent" signal. A search over all 2,120 affine lattice automorphisms of Z_15 x Z_14, crossed with monomial shifts and sector exchange, found no isomorphism between them; that is evidence they are distinct, not proof, since WL equality is not a statement either way about equality.

Only one is filed. They share (n, k, d) and therefore a slug, and filing a WL-flagged pair as two independent records would overstate what is known. The second presentation is left unfiled rather than filed as a -b twin.

Dead end worth recording: the paper's [[432,4,10]] exact row is dominated eleven times over; [[180,4,14]] and [[140,6,10]] from the same table are already on this board and are themselves dominated by [[168,4,14]].

Tools

Human-directed agent work (opencode CLI) rebuilding and screening published constructions; research/kit/bb.py for construction, research/kit/surrogate.py and the gf2_fast accelerator for the witness search, verify/validate_candidate.py for the gate. Roughly 20 CPU-minutes of witness search on an n=420 code.

Reproduction

import sys; sys.path[:0] = ["research/kit"]
from bb import build_bb
HX, HZ = build_bb(15, 14, [(3, 1), (0, 0)], [(10, 6), (11, 2), (0, 0)])

Terms are (x, y) exponent pairs against F_2[x,y]/(x^15 - 1, y^14 - 1); H_X = [A|B], H_Z = [B^T|A^T] per research/kit/bb.py. Source of the presentation: arXiv:2610.06623v1, Table 3. Construction by Juan Cruz-Benito (@cbjuan); the reconstruction and the distance witnesses are by @MathysRennela.

Parity checks

X-checks 210 (max weight 5) · Z-checks 210 (max weight 5)

H_X (210 checks, sparse supports)

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

H_Z (210 checks, sparse supports)

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

Code ID 420-4-22 · download JSON · raw on GitHub