← back to the board
[[126,24,11]] d ≤
n
126
k
24
d
11
kd²/n
23.048
w
9
X/Z
1

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 11, d_Z ≤ 11 · 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 11 · witness weight 11 (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, 11 qubits)
[10, 15, 20, 23, 38, 43, 50, 77, 80, 81, 91]
d_Z 11 · witness weight 11 (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, 11 qubits)
[33, 48, 50, 52, 71, 81, 84, 98, 110, 121, 125]
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)×252 (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): 252 (3,7): 1260 (3,8): 4284 (3,9): 18732 (3,10): 24948 (3,11): 8694 (3,12): 2520 (3,13): 630
trapping sets H_Z (1,4)×63 (2,6)×378 (3,6)×252 (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): 252 (3,7): 1260 (3,8): 4284 (3,9): 18732 (3,10): 24948 (3,11): 8694 (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, 9, 22, 52}, b = {0, 4, 21, 48} 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

[[126,24,11]] — 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/126-24-11.json.

Parity checks

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