[[420,8,13]] · QEC Challenge

[[420,8,13]] d ≤

420

kd²/n

3.219

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 13, d_Z ≤ 13 · 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 13 · witness weight 13 (claimed upper_bound)

witness operator (support, 13 qubits)

[34, 51, 68, 76, 107, 132, 149, 188, 286, 289, 292, 300, 373]

d_Z 13 · witness weight 13 (claimed upper_bound)

witness operator (support, 13 qubits)

[70, 143, 154, 157, 165, 255, 294, 311, 336, 367, 389, 392, 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

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, lattice (15,14), taken from Table 3 of arXiv:2610.06623v1 (class CL-4a95cc39). The paper reports this presentation with a decisive upper endpoint supported only by a stored BP-OSD-family scalar (no persisted witness) and no exact certificate. The distance claimed here is a witness-backed upper bound from this repository's own tightening audit (scripts/audit_weight5_css_upper_bound_witnesses.py), not the paper's raw decoder scalar. Construction by Juan Cruz-Benito (@cbjuan); the witness search is this repository's own checked-in audit artifact.

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-4a95cc39. This entry is an independent reconstruction and re-measurement, not a new construction. An independently rebuilt and re-verified weight-13 logical operator (zero syndrome, outside the stabilizer row space, re-confirmed via a second, algorithmically independent GF(2) implementation) replaces the paper's own raw BP-OSD scalar at this row. The paper prints a second, BLISS-confirmed non-isomorphic (15,14) presentation at the same [[420,8,13]] point (class CL-d192c3d1, A = 1 + y^5, B = 1 + x^6 + x^8 y^5), also witness-hardened to d<=13 by the same audit; it is not filed as a separate board entry since it would not advance any new frontier point, matching how this project's already-merged [[420,4,22]] entry handled its own second class.

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

[[420,8,13]] — 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 row, the paper's class CL-4a95cc39, originally carried a decisive upper endpoint supported only by a stored BP-OSD-family scalar (no persisted witness) — the single most failure-prone evidence tier the author's own prior tracking has seen on this catalogue (most audited rows of that tier lost 30-60% of their claimed distance once an independent search was run against them). A prior, private hardening pass already replaced that scalar with an explicit, independently reconstructed and re-verified weight-13 logical operator on each side. This submission carries that already-hardened result to the board, re-deriving the check matrices from scratch in this PR rather than reusing anything not committed here.

What was searched

No fresh search for this PR. Direct re-verification of one already-hardened, already-published presentation: l = 15, m = 14, A = 1 + y^3, B = 1 + x y^3 + x^4, so n = 2*15*14 = 420.

The check matrices were rebuilt from these 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 originally recorded only a raw decoder scalar

as the upper bound for this row (lower bound d >= 5 from exhaustive weight-four exclusion). A prior, private 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) found an explicit weight-13 logical operator on each side: zero syndrome against the opposite check matrix, outside the stabilizer row space, re-confirmed by a second, separately coded exclusion check.

This PR independently re-derives H_X/H_Z from the published

polynomials from scratch with research/kit/bb.py, re-checks both witnesses' commutation (zero syndrome) and row-space exclusion directly against these freshly rebuilt matrices, and re-confirms k = 8 — a clean rebuild against this repository's own construction tool, not a copy of unreviewed matrices.

The paper prints a second, BLISS-confirmed non-isomorphic (15,14)

presentation at the same [[420,8,13]] point (class CL-d192c3d1, A = 1 + y^5, B = 1 + x^6 + x^8 y^5), hardened to d <= 13 by the same prior pass. It is not filed as a separate board entry here: it ties this entry's point exactly and would not advance any new frontier cell, matching how this project's already-merged [[420,4,22]] entry handled its own second (possibly-isomorphic) class.

Submitted as a witness-backed upper bound, d <= 13, on both sides.

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

uv run python verify/qldpc_verify.py codes/420-8-13.json passes locally

with ok: true, earned_distance.d = {value: 13, 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

None for this row directly in this PR — the hardening work (collapsing the raw decoder scalar to a verified witness) was done in a prior private pass, not here. That pass's own history is worth recording: an early version of the same search combined two logical-flagged rows from the same coset without re-checking that their XOR was still outside the stabilizer row space, which can produce a spuriously light invalid candidate when the two rows are representatives of the same logical coset. The version that produced this witness re-verifies every combined candidate before accepting it, and this PR re-verifies the final result again from scratch.

Tools

Human-directed agent work (Claude Code, Claude Opus/Sonnet 5) rebuilding and re-verifying an already-hardened, published construction, using research/kit/bb.py for construction and verify/qldpc_verify.py for the local gate. The witness itself was produced in an earlier, private pass by the randomized kernel/permutation method described above; the method is specified in full so it can be rewritten directly against research/kit/bb.py's matrices.

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, 0), (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-4a95cc39. Construction, the witness-hardening pass 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, 266] [1, 4, 211, 228, 267] [2, 5, 212, 229, 268] [3, 6, 213, 230, 269] [4, 7, 214, 231, 270] [5, 8, 215, 232, 271] [6, 9, 216, 233, 272] [7, 10, 217, 234, 273] [8, 11, 218, 235, 274] [9, 12, 219, 236, 275] [10, 13, 220, 237, 276] [0, 11, 221, 224, 277] [1, 12, 222, 225, 278] [2, 13, 223, 226, 279] [14, 17, 224, 241, 280] [15, 18, 225, 242, 281] [16, 19, 226, 243, 282] [17, 20, 227, 244, 283] [18, 21, 228, 245, 284] [19, 22, 229, 246, 285] [20, 23, 230, 247, 286] [21, 24, 231, 248, 287] [22, 25, 232, 249, 288] [23, 26, 233, 250, 289] [24, 27, 234, 251, 290] [14, 25, 235, 238, 291] [15, 26, 236, 239, 292] [16, 27, 237, 240, 293] [28, 31, 238, 255, 294] [29, 32, 239, 256, 295] [30, 33, 240, 257, 296] [31, 34, 241, 258, 297] [32, 35, 242, 259, 298] [33, 36, 243, 260, 299] [34, 37, 244, 261, 300] [35, 38, 245, 262, 301] [36, 39, 246, 263, 302] [37, 40, 247, 264, 303] [38, 41, 248, 265, 304] [28, 39, 249, 252, 305] [29, 40, 250, 253, 306] [30, 41, 251, 254, 307] [42, 45, 252, 269, 308] [43, 46, 253, 270, 309] [44, 47, 254, 271, 310] [45, 48, 255, 272, 311] [46, 49, 256, 273, 312] [47, 50, 257, 274, 313] [48, 51, 258, 275, 314] [49, 52, 259, 276, 315] [50, 53, 260, 277, 316] [51, 54, 261, 278, 317] [52, 55, 262, 279, 318] [42, 53, 263, 266, 319] [43, 54, 264, 267, 320] [44, 55, 265, 268, 321] [56, 59, 266, 283, 322] [57, 60, 267, 284, 323] [58, 61, 268, 285, 324] [59, 62, 269, 286, 325] [60, 63, 270, 287, 326] [61, 64, 271, 288, 327] [62, 65, 272, 289, 328] [63, 66, 273, 290, 329] [64, 67, 274, 291, 330] [65, 68, 275, 292, 331] [66, 69, 276, 293, 332] [56, 67, 277, 280, 333] [57, 68, 278, 281, 334] [58, 69, 279, 282, 335] [70, 73, 280, 297, 336] [71, 74, 281, 298, 337] [72, 75, 282, 299, 338] [73, 76, 283, 300, 339] [74, 77, 284, 301, 340] [75, 78, 285, 302, 341] [76, 79, 286, 303, 342] [77, 80, 287, 304, 343] [78, 81, 288, 305, 344] [79, 82, 289, 306, 345] [80, 83, 290, 307, 346] [70, 81, 291, 294, 347] [71, 82, 292, 295, 348] [72, 83, 293, 296, 349] [84, 87, 294, 311, 350] [85, 88, 295, 312, 351] [86, 89, 296, 313, 352] [87, 90, 297, 314, 353] [88, 91, 298, 315, 354] [89, 92, 299, 316, 355] [90, 93, 300, 317, 356] [91, 94, 301, 318, 357] [92, 95, 302, 319, 358] [93, 96, 303, 320, 359] [94, 97, 304, 321, 360] [84, 95, 305, 308, 361] [85, 96, 306, 309, 362] [86, 97, 307, 310, 363] [98, 101, 308, 325, 364] [99, 102, 309, 326, 365] [100, 103, 310, 327, 366] [101, 104, 311, 328, 367] [102, 105, 312, 329, 368] [103, 106, 313, 330, 369] [104, 107, 314, 331, 370] [105, 108, 315, 332, 371] [106, 109, 316, 333, 372] [107, 110, 317, 334, 373] [108, 111, 318, 335, 374] [98, 109, 319, 322, 375] [99, 110, 320, 323, 376] [100, 111, 321, 324, 377] [112, 115, 322, 339, 378] [113, 116, 323, 340, 379] [114, 117, 324, 341, 380] [115, 118, 325, 342, 381] [116, 119, 326, 343, 382] [117, 120, 327, 344, 383] [118, 121, 328, 345, 384] [119, 122, 329, 346, 385] [120, 123, 330, 347, 386] [121, 124, 331, 348, 387] [122, 125, 332, 349, 388] [112, 123, 333, 336, 389] [113, 124, 334, 337, 390] [114, 125, 335, 338, 391] [126, 129, 336, 353, 392] [127, 130, 337, 354, 393] [128, 131, 338, 355, 394] [129, 132, 339, 356, 395] [130, 133, 340, 357, 396] [131, 134, 341, 358, 397] [132, 135, 342, 359, 398] [133, 136, 343, 360, 399] [134, 137, 344, 361, 400] [135, 138, 345, 362, 401] [136, 139, 346, 363, 402] [126, 137, 347, 350, 403] [127, 138, 348, 351, 404] [128, 139, 349, 352, 405] [140, 143, 350, 367, 406] [141, 144, 351, 368, 407] [142, 145, 352, 369, 408] [143, 146, 353, 370, 409] [144, 147, 354, 371, 410] [145, 148, 355, 372, 411] [146, 149, 356, 373, 412] [147, 150, 357, 374, 413] [148, 151, 358, 375, 414] [149, 152, 359, 376, 415] [150, 153, 360, 377, 416] [140, 151, 361, 364, 417] [141, 152, 362, 365, 418] [142, 153, 363, 366, 419] [154, 157, 210, 364, 381] [155, 158, 211, 365, 382] [156, 159, 212, 366, 383] [157, 160, 213, 367, 384] [158, 161, 214, 368, 385] [159, 162, 215, 369, 386] [160, 163, 216, 370, 387] [161, 164, 217, 371, 388] [162, 165, 218, 372, 389] [163, 166, 219, 373, 390] [164, 167, 220, 374, 391] [154, 165, 221, 375, 378] [155, 166, 222, 376, 379] [156, 167, 223, 377, 380] [168, 171, 224, 378, 395] [169, 172, 225, 379, 396] [170, 173, 226, 380, 397] [171, 174, 227, 381, 398] [172, 175, 228, 382, 399] [173, 176, 229, 383, 400] [174, 177, 230, 384, 401] [175, 178, 231, 385, 402] [176, 179, 232, 386, 403] [177, 180, 233, 387, 404] [178, 181, 234, 388, 405] [168, 179, 235, 389, 392] [169, 180, 236, 390, 393] [170, 181, 237, 391, 394] [182, 185, 238, 392, 409] [183, 186, 239, 393, 410] [184, 187, 240, 394, 411] [185, 188, 241, 395, 412] [186, 189, 242, 396, 413] [187, 190, 243, 397, 414] [188, 191, 244, 398, 415] [189, 192, 245, 399, 416] [190, 193, 246, 400, 417] [191, 194, 247, 401, 418] [192, 195, 248, 402, 419] [182, 193, 249, 403, 406] [183, 194, 250, 404, 407] [184, 195, 251, 405, 408] [196, 199, 213, 252, 406] [197, 200, 214, 253, 407] [198, 201, 215, 254, 408] [199, 202, 216, 255, 409] [200, 203, 217, 256, 410] [201, 204, 218, 257, 411] [202, 205, 219, 258, 412] [203, 206, 220, 259, 413] [204, 207, 221, 260, 414] [205, 208, 222, 261, 415] [206, 209, 223, 262, 416] [196, 207, 210, 263, 417] [197, 208, 211, 264, 418] [198, 209, 212, 265, 419]

H_Z (210 checks, sparse supports)

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

Code ID 420-8-13 · download JSON · raw on GitHub