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[[126,18,13]] d ≤
n
126
k
18
d
13
kd²/n
24.143
w
9
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 13, d_Z ≤ 13 · w_X = 9, w_Z = 9 (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 found by @mathysrennela · research/kit/surrogate.lightest_logical · found at 2000 trials · survived 2000 trials · 2026-10-12
witness operator (support, 13 qubits)
[11, 17, 31, 53, 60, 66, 83, 88, 94, 107, 110, 113, 119]
d_Z 13 · witness weight 13 (claimed upper_bound)
witness found by @mathysrennela · research/kit/surrogate.lightest_logical · found at 2000 trials · survived 2000 trials · 2026-10-12
witness operator (support, 13 qubits)
[2, 6, 8, 18, 32, 46, 60, 62, 67, 69, 81, 82, 89]
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 9 · H_Z 9
qubit degrees H_X 4–5 (mean 4.5) · H_Z 4–5 (mean 4.5)
trapping sets H_X (1,4)×63 (2,6)×378 (3,6)×315 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 63 (1,5): 63 (2,6): 378 (2,7): 1260 (2,8): 630 (3,6): 315 (3,7): 1260 (3,8): 4536 (3,9): 18585 (3,10): 23625 (3,11): 9135 (3,12): 2520 (3,13): 630
trapping sets H_Z (1,4)×63 (2,6)×378 (3,6)×315 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 63 (1,5): 63 (2,6): 378 (2,7): 1260 (2,8): 630 (3,6): 315 (3,7): 1260 (3,8): 4536 (3,9): 18585 (3,10): 23625 (3,11): 9135 (3,12): 2520 (3,13): 630

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C63 = SmallGroup(63, 2) (order 63), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 126; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 2, 11, 49, 58}, b = {0, 5, 51, 55} as element indices. Check weight 9. Parameters and blocks taken from the public code database of arXiv:2610.13880v1 (Lee, 'Bicycle Race'), github.com/gideonlsx/bicycle-race.
model Mimo-V2.6-Flash (claimed, not verified)
reference arXiv:2610.13880
date 2026-10-12
notes Parameters and construction taken from the public database of arXiv:2610.13880v1. Distance is an upper bound certified by the witnesses attached here. Parameters and blocks taken from the public code database of arXiv:2610.13880v1 (Lee, 'Bicycle Race'), github.com/gideonlsx/bicycle-race @ aed705a, CC BY 4.0; rebuilt here with the repo's own group data. Distance is an upper bound certified by the attached witnesses, not the paper's exact distance.
family generalized bicycle (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight > 8 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[126,18,13]] — two-block group-algebra code over C63

Direction & hypothesis

Cell weight-9plus x unrestricted, CSS board. The argument is coverage, not discovery: this code is published in arXiv:2610.13880v1 (Lee, "Bicycle Race"), and nothing on the board occupied this (n, k, d, w) point. It advances the four-axis Pareto frontier by moving into a cell that was empty at these parameters.

What was searched

Not a search for new codes. The arXiv:2610.13880v1 database (github.com/gideonlsx/bicycle-race @ aed705a, CC BY 4.0, 12,448 records) was rebuilt here from its group and block data, then screened against every CSS entry on main under the site's Pareto rule (n <=, k >=, d >=, w <= with at least one strict) within the same weight class. Records whose parameters the paper already tabulates, and candidates claimed by other open filings, were removed first. Distances were re-searched locally:

  • X side: research/kit/surrogate.lightest_logical at 2,000 trials
  • Z side: research/kit/surrogate.lightest_logical at 2,000 trials

Evidence trail

Witness-backed upper bound, not an exact distance. The attached X and Z operators were found by the searches above and re-checked by the verifier's own criteria (in ker(opposite) and outside rowspace(own)). verify/validate_candidate.py reports passed: true and board_advancing: true, with d_only_gain: false.

The paper states an exact distance for this record, verified there with a second exact engine plus GAP rebuild and independent upper-bound attacks. That exactness is not claimed here; the board records what the witness certifies.

Dead ends

  • Direct sums of two board codes: rejected, CONTRIBUTING.md requires a single

    • connected Tanner graph and a connected stabilizer group.

  • Fitting a single-layer 2D layout to reach the local-2d cells: searched with

    • research/local2d/fold_layout.search_layout, best radii 10–13 against a ceiling of 4.0, so these codes stay unrestricted.

  • Hypergraph products of two parents from this pool: connectivity passes, but n

    • multiplies (two ~250-qubit parents give n = 80,010).

Tools

Model: Mimo-V2.6-Flash (self-reported, provenance.model), driving OpenCode. Rebuild: research/kit/group_algebra.build_2bga from the database's data/groups.jsonl Cayley table. Witness search: research/kit/surrogate (numpy backend, 2,000 trials/side screening) with the gf2_fast accelerator for longer runs. Gate: research/kit/coordination.py gate, which runs verify/validate_candidate.py unchanged. Search run by @mathysrennela.

Reproduction

from group_algebra import build_2bga
# a and b are element indices from github.com/gideonlsx/bicycle-race @ aed705a data/codes.jsonl
HX, HZ = build_2bga(mul, a, b)   # mul from data/groups.jsonl, n = 2|G| = 126

Parameters are also spelled out in provenance.construction of codes/126-18-13.json.

Parity checks

X-checks 63 (max weight 9) · Z-checks 63 (max weight 9)
H_X (63 checks, sparse supports)
[0, 6, 8, 28, 42, 63, 80, 84, 125] [1, 11, 13, 26, 48, 64, 86, 90, 105] [0, 2, 13, 16, 37, 65, 83, 89, 93] [3, 14, 17, 38, 51, 65, 66, 94, 116] [4, 9, 19, 32, 53, 67, 88, 92, 111] [1, 5, 19, 22, 35, 63, 68, 95, 99] [6, 7, 20, 23, 56, 68, 69, 100, 114] [2, 7, 22, 25, 46, 70, 92, 98, 102] [3, 8, 23, 26, 47, 67, 70, 71, 103] [1, 2, 9, 18, 58, 71, 72, 104, 122] [4, 10, 17, 28, 41, 64, 73, 97, 101] [0, 11, 12, 29, 59, 73, 74, 98, 119] [5, 12, 28, 31, 44, 65, 75, 104, 108] [6, 13, 15, 29, 32, 66, 75, 76, 109] [4, 5, 14, 16, 61, 76, 77, 110, 121] [7, 15, 31, 34, 54, 78, 101, 107, 111] [8, 16, 32, 35, 55, 73, 78, 79, 112] [5, 7, 9, 17, 27, 74, 79, 80, 113] [10, 18, 26, 37, 50, 68, 81, 106, 110] [2, 11, 19, 21, 38, 69, 81, 82, 107] [3, 10, 20, 22, 62, 78, 82, 83, 124] [12, 21, 37, 40, 52, 70, 84, 113, 116] [13, 22, 24, 38, 41, 71, 84, 85, 117] [10, 12, 14, 23, 25, 72, 85, 86, 118] [15, 24, 40, 43, 59, 64, 87, 110, 115] [16, 25, 41, 44, 60, 81, 87, 88, 119] [12, 15, 17, 26, 36, 82, 88, 89, 120] [18, 27, 35, 46, 57, 75, 90, 114, 118] [7, 19, 28, 30, 47, 76, 90, 91, 115] [8, 18, 20, 29, 31, 77, 87, 91, 92] [21, 30, 46, 49, 58, 67, 78, 93, 120] [22, 31, 33, 47, 50, 79, 93, 94, 122] [18, 21, 23, 32, 34, 80, 94, 95, 123] [11, 24, 33, 49, 51, 68, 96, 118, 121] [25, 34, 50, 52, 62, 69, 90, 96, 97] [21, 24, 26, 35, 45, 91, 97, 98, 124] [27, 36, 44, 54, 61, 66, 84, 99, 123] [15, 28, 37, 39, 55, 85, 99, 100, 121] [16, 27, 29, 38, 40, 86, 96, 100, 101] [9, 30, 39, 54, 56, 73, 87, 102, 124] [31, 40, 42, 55, 57, 74, 88, 102, 103] [27, 30, 32, 41, 43, 89, 103, 104, 125] [3, 19, 33, 42, 56, 72, 75, 105, 123] [20, 34, 43, 57, 58, 76, 99, 105, 106] [30, 33, 35, 44, 53, 77, 100, 106, 107] [14, 36, 45, 52, 59, 71, 93, 108, 125] [24, 37, 46, 48, 60, 72, 94, 108, 109] [25, 36, 38, 47, 49, 95, 105, 109, 110] [6, 17, 39, 48, 59, 77, 81, 96, 111] [0, 40, 49, 60, 61, 82, 97, 111, 112] [36, 39, 41, 50, 51, 83, 98, 112, 113] [9, 29, 43, 51, 61, 63, 85, 108, 114] [4, 39, 42, 44, 52, 86, 109, 114, 115] [11, 23, 45, 53, 58, 79, 83, 102, 116] [1, 33, 46, 54, 62, 80, 103, 116, 117] [34, 45, 47, 55, 56, 63, 104, 117, 118] [2, 14, 49, 56, 62, 64, 91, 106, 119] [3, 45, 48, 50, 57, 92, 107, 119, 120] [0, 10, 48, 52, 58, 66, 95, 117, 121] [5, 20, 42, 54, 59, 67, 89, 112, 122] [6, 43, 53, 55, 60, 65, 113, 122, 123] [1, 8, 53, 57, 61, 69, 101, 115, 124] [4, 13, 51, 60, 62, 70, 74, 120, 125]
H_Z (63 checks, sparse supports)
[0, 5, 51, 55, 63, 65, 74, 112, 121] [1, 10, 24, 56, 64, 68, 72, 117, 124] [2, 3, 12, 60, 65, 70, 72, 82, 119] [3, 13, 36, 58, 66, 71, 83, 105, 120] [4, 8, 30, 59, 67, 73, 77, 115, 125] [5, 6, 18, 33, 68, 75, 77, 80, 122] [6, 19, 34, 61, 63, 69, 76, 111, 123] [7, 8, 21, 62, 69, 70, 78, 80, 91] [8, 9, 22, 45, 63, 71, 79, 92, 124] [9, 23, 42, 46, 67, 72, 80, 102, 114] [10, 11, 16, 39, 73, 81, 83, 86, 121] [11, 17, 40, 62, 64, 74, 82, 96, 116] [12, 13, 27, 42, 74, 75, 84, 86, 89] [13, 14, 28, 43, 64, 65, 76, 85, 125] [14, 29, 44, 48, 66, 77, 86, 108, 119] [15, 16, 20, 30, 76, 78, 87, 89, 100] [16, 17, 31, 53, 65, 77, 79, 88, 101] [0, 17, 32, 54, 66, 73, 80, 89, 111] [18, 19, 25, 48, 72, 81, 90, 92, 95] [19, 20, 26, 49, 67, 68, 82, 91, 105] [2, 20, 50, 53, 69, 83, 92, 106, 122] [0, 21, 22, 36, 82, 84, 93, 95, 98] [22, 23, 37, 51, 68, 70, 83, 85, 94] [1, 23, 38, 52, 69, 71, 86, 95, 116] [24, 25, 29, 39, 85, 87, 96, 98, 109] [4, 25, 26, 40, 70, 86, 88, 97, 110] [2, 26, 41, 59, 64, 71, 81, 89, 98] [1, 27, 28, 34, 80, 90, 99, 101, 104] [28, 29, 35, 56, 63, 73, 75, 91, 100] [4, 7, 29, 57, 74, 76, 92, 101, 114] [2, 30, 31, 45, 91, 93, 102, 104, 107] [3, 31, 32, 46, 75, 78, 92, 94, 103] [5, 32, 47, 58, 67, 76, 79, 95, 104] [33, 34, 38, 48, 94, 96, 105, 107, 117] [10, 34, 35, 49, 78, 95, 97, 106, 118] [7, 11, 35, 50, 68, 79, 90, 98, 107] [5, 36, 37, 43, 89, 99, 108, 110, 113] [6, 37, 38, 44, 65, 81, 84, 100, 109] [10, 15, 38, 61, 66, 82, 85, 101, 110] [7, 39, 40, 53, 100, 102, 111, 113, 115] [8, 40, 41, 54, 84, 87, 101, 103, 112] [9, 12, 41, 55, 73, 85, 88, 104, 113] [1, 42, 43, 47, 63, 103, 105, 115, 122] [18, 43, 44, 56, 87, 104, 106, 114, 123] [15, 19, 44, 57, 75, 88, 99, 107, 115] [12, 45, 46, 51, 98, 108, 116, 118, 120] [13, 46, 47, 52, 70, 90, 93, 109, 117] [14, 18, 24, 47, 71, 91, 94, 110, 118] [4, 15, 48, 49, 64, 109, 111, 120, 121] [16, 49, 50, 59, 93, 96, 110, 112, 119] [17, 21, 50, 60, 81, 94, 97, 113, 120] [6, 27, 51, 52, 66, 96, 113, 114, 125] [24, 28, 52, 61, 84, 97, 108, 115, 121] [3, 21, 53, 54, 67, 107, 116, 123, 124] [22, 54, 55, 58, 78, 99, 102, 117, 122] [23, 27, 33, 55, 79, 100, 103, 118, 123] [11, 25, 56, 57, 69, 102, 105, 118, 119] [26, 30, 57, 62, 90, 103, 106, 120, 124] [14, 33, 37, 58, 72, 93, 106, 116, 121] [9, 31, 59, 60, 74, 87, 108, 111, 122] [32, 36, 42, 60, 88, 109, 112, 123, 125] [20, 35, 39, 61, 77, 99, 112, 114, 124] [0, 41, 45, 62, 83, 97, 117, 119, 125]
Code ID 126-18-13 · download JSON · raw on GitHub