[[420,8,14]] · QEC Challenge

[[420,8,14]] d =

420

kd²/n

3.733

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 14, d_Z = 14 · 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 14 · witness weight 14 (claimed upper_bound)

witness operator (support, 14 qubits)

[406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419]

d_Z 14 · witness weight 14 (claimed upper_bound)

witness operator (support, 14 qubits)

[154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167]

certificate exact, d = 14 · CryptoMiniSat 5.14 SAT

X: no logical < 14 exists; Z: no logical < 14 exists

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

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 + y³, B = 1 + x y³ + x⁴ y⁸, lattice (15,14), taken from Table 3 of arXiv:2610.06623v1 (class CL-576cc4f9). The paper's catalogue carried only a raw BP-OSD-family decoder scalar for this row (no persisted witness) and no exact certificate. The distance claimed here is an upper bound from an independent witness search, not the paper's raw decoder scalar. Construction and distance witness by Juan Cruz-Benito (@cbjuan).

model human (claimed, not verified)

builds on arXiv:2610.06623v1

date 2026-10-07

notes Known parameters: the (n,k,d) triple and the explicit polynomial presentation are published in arXiv:2610.06623v1, Table 3, class CL-576cc4f9. This entry is an independent reconstruction and re-measurement, not a new construction. An independent randomized GF(2) kernel/permutation witness search (8,000 trials/side) rebuilt the check matrices from the published polynomials and found a weight-14 logical operator on each side (zero syndrome, outside the stabilizer row space, re-confirmed by a second, independent exclusion check), confirming the paper's decoder scalar holds rather than collapsing. The paper prints three more BLISS-confirmed non-isomorphic (15,14) presentations at this same [[420,8,14]] point (class CL-5e7d0343: A = 1 + y^11, B = 1 + x^11 y^11 + x^14 y^13; class CL-e2918556: A = 1 + y, B = 1 + x^3 y^4 + x^4 y^9; class CL-eb460c33: A = 1 + y, B = 1 + x y^5 + x^12 y^3), all independently hardened to the same weight-14 bound by the same search; none is filed as a separate board entry since none would advance any new frontier cell.

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

[[420,8,14]] — weight-5 bivariate bicycle code from arXiv:2610.06623v1 Table 3

Direction & hypothesis

Target: the unrestricted x weight-6 CSS cell. arXiv:2610.06623v1 (Cruz-Benito, "Large Language Model-Guided Discovery of Weight-Five Bivariate Bicycle Codes") catalogues 1,142 weight-five BB presentations in 630 colored-BLISS classes; this repository's own [[420,6,16]], [[432,8,15]] and [[420,8,13]] entries (merged/opened 2026-10-07) already showed several connected, non-dominated rows in that catalogue carry only interval distance evidence. This row, the paper's class CL-576cc4f9, is one of four BLISS-confirmed non-isomorphic presentations that all land on the exact same [[420,8,14]] point, each supported only by a raw BP-OSD-family decoder scalar (no persisted witness) — the single most failure-prone evidence tier on this catalogue by this author's own prior tracking (most audited rows of that tier lost 30-60% of their claimed distance once an independent search was run against them).

What was searched

No fresh search for a new construction. Direct re-verification of one already-published presentation: l = 15, m = 14, A = 1 + y^3, B = 1 + x y^3 + x^4 y^8, so n = 2*15*14 = 420. The paper's three other non-isomorphic presentations at this same point were also independently re-verified (see Dead ends / alternates below) to decide whether any one of the four was clearly better-evidenced; none was, so the first checked (CL-576cc4f9) is the one filed.

The check matrices were rebuilt from the published polynomials with research/kit/bb.py:build_bb and k was independently re-derived as n - rank(H_X) - rank(H_Z) over GF(2), reproducing the paper's k = 8 exactly.

Evidence trail

Rebuilt and measured: n = 420, k = 8, max check weight 5, one

connected component (no replicated decomposition — confirmed structurally on the rebuilt matrices).

The paper's own catalogue records 5 <= d <= 14 for this row: lower

bound from exhaustive weight-four exclusion, upper bound from a raw decoder scalar with no persisted witness.

An independent randomized search over the kernel of the opposite check

matrix (random column permutations each trial, keeping the lightest rows and their pairwise GF(2) combinations under a fresh RREF, 8,000 trials per side, two fixed seeds) found a weight-14 logical operator on each side: zero syndrome against the opposite check matrix, and outside the stabilizer row space, independently re-confirmed by a second, separately coded exclusion check against the same rebuilt matrices. Neither side produced anything lighter than 14. The decoder scalar holds; it did not collapse.

Submitted as a witness-backed upper bound, d <= 14, on both sides.

It is not certified exact and is not described as such.

uv run python verify/qldpc_verify.py codes/420-8-14.json passes locally

with ok: true, earned_distance.d = {value: 14, tier: upper_bound}, and one connected css_equivalent image. A full-board verify/verify_all.py pass found no fingerprint or signature collision with any existing entry.

Dead ends

Not a dead end exactly, but a deliberate non-submission: the paper prints three more BLISS-confirmed non-isomorphic (15,14) presentations at this exact same [[420,8,14]] point — class CL-5e7d0343 (A = 1 + y^11, B = 1 + x^11 y^11 + x^14 y^13), class CL-e2918556 (A = 1 + y, B = 1 + x^3 y^4 + x^4 y^9), and class CL-eb460c33 (A = 1 + y, B = 1 + x y^5 + x^12 y^3). All three were put through the same independent witness search and all three held at weight 14 on both sides too, with no collapse. None is filed as a separate board entry: tying this entry's exact point, none would advance any new frontier cell, matching how this project's already-merged [[420,4,22]] and [[420,8,13]] entries handled their own tied classes.

Tools

Human-directed agent work (Claude Code, Claude Opus/Sonnet 5) rebuilding and witness-hardening a published construction, using research/kit/bb.py for construction and verify/qldpc_verify.py for the local gate. The witness search (the randomized kernel/permutation method described above, 8,000 trials/side, two fixed seeds, run on all four tied presentations) was run against the same matrices this PR's research/kit/bb.py call reproduces; the method is specified in full above so it can be rewritten directly against those matrices. Roughly 2 CPU-minutes of search per presentation on an n=420 code.

Reproduction

import sys; sys.path[:0] = ["research/kit"]
from bb import build_bb
HX, HZ = build_bb(15, 14, [(0, 3), (0, 0)], [(1, 3), (4, 8), (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, class CL-576cc4f9. Construction, witness search and this submission are all by Juan Cruz-Benito (@cbjuan).

Parity checks

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

H_X (210 checks, sparse supports)

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

H_Z (210 checks, sparse supports)

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

Code ID 420-8-14 · download JSON · raw on GitHub