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[[120,20,11]] d ≤
n
120
k
20
d
11
kd²/n
20.167
w
9
X/Z
1

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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)
[23, 42, 47, 48, 50, 74, 76, 77, 81, 88, 107]
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)
[1, 11, 12, 24, 29, 34, 49, 57, 61, 94, 112]
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)×60 (2,6)×480 (3,6)×660 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 60 (1,5): 60 (2,6): 480 (2,7): 1200 (2,8): 360 (3,6): 660 (3,7): 1080 (3,8): 5880 (3,9): 20040 (3,10): 17220 (3,11): 4800 (3,12): 1440 (3,13): 120
trapping sets H_Z (1,4)×60 (2,6)×480 (3,6)×660 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 60 (1,5): 60 (2,6): 480 (2,7): 1200 (2,8): 360 (3,6): 660 (3,7): 1080 (3,8): 5880 (3,9): 20040 (3,10): 17220 (3,11): 4800 (3,12): 1440 (3,13): 120

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C30 x C2 = SmallGroup(60, 13) (order 60), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 120; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 8, 36, 37, 46}, b = {0, 14, 39, 59} 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

[[120,20,11]] — two-block group-algebra code over C30 x C2

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| = 120

Parameters are also spelled out in provenance.construction of codes/120-20-11.json.

Parity checks

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