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[[124,10,15]] d ≤
n
124
k
10
d
15
kd²/n
18.145
w
9
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 15, d_Z ≤ 15 · 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 15 · witness weight 15 (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, 15 qubits)
[9, 10, 25, 30, 35, 38, 52, 53, 59, 63, 76, 77, 81, 94, 104]
d_Z 15 · witness weight 15 (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, 15 qubits)
[17, 20, 21, 25, 27, 33, 44, 58, 62, 70, 76, 77, 81, 90, 102]
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 4 · H_Z 4 (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)×62 (2,5)×62 (3,6)×496 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 62 (1,5): 62 (2,5): 62 (2,6): 372 (2,7): 1116 (2,8): 620 (3,6): 496 (3,7): 2046 (3,8): 4960 (3,9): 16740 (3,10): 21266 (3,11): 8060 (3,12): 1984 (3,13): 620
trapping sets H_Z (1,4)×62 (2,5)×62 (3,6)×496 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 62 (1,5): 62 (2,5): 62 (2,6): 372 (2,7): 1116 (2,8): 620 (3,6): 496 (3,7): 2046 (3,8): 4960 (3,9): 16740 (3,10): 21266 (3,11): 8060 (3,12): 1984 (3,13): 620

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C62 = SmallGroup(62, 2) (order 62), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 124; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 4, 12, 25, 56}, b = {0, 3, 24, 50} 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

[[124,10,15]] — two-block group-algebra code over C62

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| = 124

Parameters are also spelled out in provenance.construction of codes/124-10-15.json.

Parity checks

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