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[[126,20,7]] d ≤
n
126
k
20
d
7
kd²/n
7.778
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 7, d_Z ≤ 7 · w_X = 7, w_Z = 7 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 7 · witness weight 7 (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, 7 qubits)
[27, 28, 38, 57, 65, 87, 125]
d_Z 7 · witness weight 7 (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, 7 qubits)
[23, 49, 55, 90, 100, 101, 115]
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 7 · H_Z 7
qubit degrees H_X 3–4 (mean 3.5) · H_Z 3–4 (mean 3.5)
trapping sets H_X (1,3)×63 (2,4)×189 (3,3)×63 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 63 (1,4): 63 (2,4): 189 (2,5): 756 (2,6): 378 (3,3): 63 (3,4): 63 (3,5): 882 (3,6): 6510 (3,7): 8946 (3,8): 3339 (3,9): 1134 (3,10): 252
trapping sets H_Z (1,3)×63 (2,4)×189 (3,3)×63 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 63 (1,4): 63 (2,4): 189 (2,5): 756 (2,6): 378 (3,3): 63 (3,4): 63 (3,5): 882 (3,6): 6510 (3,7): 8946 (3,8): 3339 (3,9): 1134 (3,10): 252

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C21 x C3 = SmallGroup(63, 4) (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, 5, 38, 60}, b = {0, 15, 43} as element indices. Check weight 7. 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

[[126,20,7]] — two-block group-algebra code over C21 x C3

Direction & hypothesis

Cell weight-8 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/126-20-7.json.

Parity checks

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