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[[150,8,18]] d ≤
n
150
k
8
d
18
kd²/n
17.28
w
9
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 18, d_Z ≤ 18 · 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 18 · witness weight 18 (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, 18 qubits)
[1, 5, 7, 13, 16, 33, 48, 60, 66, 67, 86, 88, 93, 110, 117, 124, 143, 147]
d_Z 18 · witness weight 18 (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, 18 qubits)
[6, 22, 23, 24, 31, 34, 38, 40, 43, 49, 57, 68, 88, 99, 108, 130, 136, 146]
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)×75 (2,6)×450 (3,6)×525 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 75 (1,5): 75 (2,6): 450 (2,7): 1500 (2,8): 750 (3,6): 525 (3,7): 900 (3,8): 4650 (3,9): 24075 (3,10): 29025 (3,11): 10425 (3,12): 3000 (3,13): 750
trapping sets H_Z (1,4)×75 (2,6)×450 (3,6)×525 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 75 (1,5): 75 (2,6): 450 (2,7): 1500 (2,8): 750 (3,6): 525 (3,7): 900 (3,8): 4650 (3,9): 24075 (3,10): 29025 (3,11): 10425 (3,12): 3000 (3,13): 750

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C75 = SmallGroup(75, 1) (order 75), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 150; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 7, 27, 28, 60}, b = {0, 11, 47, 62} 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

[[150,8,18]] — two-block group-algebra code over C75

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| = 150

Parameters are also spelled out in provenance.construction of codes/150-8-18.json.

Parity checks

X-checks 75 (max weight 9) · Z-checks 75 (max weight 9)
H_X (75 checks, sparse supports)
[0, 33, 41, 42, 64, 75, 106, 114, 124] [1, 17, 50, 51, 69, 76, 90, 123, 131] [2, 46, 53, 54, 70, 77, 89, 119, 127] [3, 15, 47, 54, 55, 76, 78, 120, 135] [4, 24, 57, 58, 72, 79, 97, 105, 115] [5, 28, 60, 61, 73, 80, 96, 101, 134] [6, 22, 29, 61, 62, 79, 81, 102, 140] [0, 7, 18, 57, 63, 82, 86, 99, 137] [8, 26, 58, 63, 64, 80, 83, 100, 131] [7, 9, 27, 59, 64, 81, 84, 132, 143] [10, 37, 65, 66, 74, 84, 85, 110, 118] [11, 31, 38, 66, 67, 75, 86, 111, 128] [1, 12, 25, 41, 68, 78, 87, 108, 142] [13, 35, 42, 68, 69, 85, 88, 109, 115] [12, 14, 36, 43, 69, 86, 89, 116, 146] [2, 15, 29, 30, 65, 90, 95, 112, 144] [3, 16, 30, 66, 70, 87, 91, 96, 113] [15, 17, 40, 67, 70, 88, 92, 114, 140] [10, 15, 16, 18, 41, 89, 93, 141, 148] [4, 19, 34, 50, 71, 81, 92, 94, 130] [20, 44, 51, 71, 72, 77, 93, 95, 124] [19, 21, 45, 52, 72, 78, 96, 125, 138] [5, 22, 38, 39, 53, 83, 97, 121, 147] [6, 23, 39, 54, 73, 84, 94, 98, 122] [22, 24, 49, 55, 73, 95, 99, 123, 128] [2, 22, 23, 25, 50, 96, 100, 129, 149] [7, 26, 34, 42, 43, 101, 107, 125, 148] [0, 8, 27, 43, 71, 97, 102, 108, 126] [0, 9, 26, 28, 72, 98, 103, 109, 127] [19, 26, 27, 29, 53, 76, 99, 104, 146] [20, 27, 28, 30, 54, 100, 105, 110, 147] [10, 31, 47, 48, 60, 88, 103, 106, 139] [11, 32, 48, 61, 74, 82, 89, 104, 107] [31, 33, 56, 62, 74, 83, 105, 108, 135] [5, 31, 32, 34, 57, 84, 109, 136, 145] [12, 18, 35, 51, 52, 91, 110, 132, 149] [1, 13, 36, 52, 63, 92, 106, 111, 133] [1, 14, 35, 37, 64, 93, 107, 112, 134] [7, 35, 36, 38, 60, 79, 108, 113, 138] [8, 36, 37, 39, 61, 109, 114, 119, 139] [2, 16, 40, 48, 55, 110, 115, 121, 136] [2, 3, 17, 41, 74, 111, 116, 122, 137] [3, 18, 31, 40, 42, 80, 112, 117, 123] [32, 40, 41, 43, 63, 81, 113, 118, 149] [19, 25, 44, 58, 59, 98, 116, 119, 145] [4, 20, 45, 59, 68, 90, 99, 117, 120] [4, 21, 44, 46, 69, 91, 100, 118, 121] [12, 44, 45, 47, 65, 75, 92, 122, 143] [13, 45, 46, 48, 66, 93, 101, 123, 144] [5, 23, 30, 49, 62, 103, 119, 124, 141] [5, 6, 24, 50, 70, 104, 120, 125, 142] [6, 15, 25, 49, 51, 85, 105, 121, 126] [16, 49, 50, 52, 68, 86, 122, 127, 145] [4, 7, 8, 28, 53, 124, 128, 133, 144] [8, 9, 29, 44, 54, 87, 125, 129, 134] [9, 30, 45, 53, 55, 79, 88, 126, 130] [10, 32, 39, 56, 67, 101, 112, 129, 131] [10, 11, 33, 57, 73, 102, 113, 130, 132] [11, 22, 34, 56, 58, 77, 103, 114, 133] [23, 56, 57, 59, 71, 78, 104, 134, 148] [0, 12, 13, 37, 60, 117, 131, 135, 147] [13, 14, 26, 38, 61, 94, 118, 132, 136] [14, 27, 39, 60, 62, 75, 95, 133, 137] [11, 16, 17, 42, 63, 97, 135, 138, 142] [17, 18, 43, 56, 64, 85, 98, 136, 139] [1, 19, 20, 46, 65, 115, 126, 139, 140] [20, 21, 35, 47, 66, 82, 116, 127, 141] [21, 36, 48, 65, 67, 76, 83, 117, 142] [3, 23, 24, 51, 68, 106, 130, 140, 143] [24, 25, 40, 52, 69, 77, 107, 141, 144] [21, 28, 29, 55, 70, 94, 111, 143, 145] [6, 32, 33, 58, 71, 90, 128, 137, 146] [33, 34, 49, 59, 72, 80, 91, 129, 147] [9, 37, 38, 62, 73, 82, 120, 146, 148] [14, 46, 47, 67, 74, 87, 102, 138, 149]
H_Z (75 checks, sparse supports)
[0, 11, 47, 62, 75, 82, 102, 103, 135] [1, 3, 29, 67, 76, 87, 111, 112, 140] [2, 20, 58, 69, 77, 90, 100, 115, 116] [3, 12, 21, 59, 78, 91, 116, 117, 143] [4, 6, 38, 55, 79, 94, 120, 121, 128] [5, 8, 42, 72, 80, 97, 109, 124, 125] [6, 9, 19, 43, 81, 98, 125, 126, 146] [7, 32, 66, 73, 82, 84, 101, 113, 128] [8, 22, 33, 67, 83, 102, 114, 128, 129] [9, 10, 23, 34, 84, 103, 129, 130, 148] [10, 13, 51, 64, 85, 93, 106, 131, 132] [7, 11, 14, 52, 86, 107, 132, 133, 138] [12, 16, 54, 74, 87, 89, 110, 122, 135] [13, 17, 31, 55, 88, 111, 123, 135, 136] [2, 14, 18, 32, 89, 112, 136, 137, 149] [1, 15, 45, 71, 78, 90, 92, 93, 126] [16, 35, 46, 72, 91, 93, 115, 127, 138] [17, 19, 36, 47, 76, 92, 116, 138, 139] [18, 20, 37, 48, 82, 93, 110, 117, 139] [19, 23, 61, 70, 94, 96, 104, 119, 140] [15, 20, 24, 62, 95, 105, 120, 140, 141] [5, 16, 21, 25, 96, 121, 141, 142, 145] [4, 22, 27, 63, 81, 97, 99, 100, 133] [23, 28, 44, 64, 98, 100, 124, 134, 143] [7, 24, 29, 45, 79, 99, 125, 143, 144] [8, 25, 30, 46, 87, 100, 119, 126, 144] [5, 26, 48, 56, 83, 101, 103, 104, 136] [6, 27, 57, 74, 84, 102, 104, 105, 137] [28, 31, 49, 58, 80, 103, 105, 128, 145] [29, 32, 50, 59, 81, 90, 104, 129, 145] [4, 30, 33, 51, 90, 91, 105, 124, 130] [0, 31, 36, 68, 86, 106, 108, 109, 117] [26, 32, 37, 69, 107, 109, 118, 131, 146] [12, 27, 33, 38, 75, 108, 132, 146, 147] [13, 28, 34, 39, 94, 101, 109, 133, 147] [10, 30, 35, 40, 88, 110, 112, 113, 141] [11, 36, 41, 70, 89, 111, 113, 114, 142] [15, 37, 42, 56, 85, 112, 114, 135, 148] [16, 38, 43, 57, 86, 97, 113, 136, 148] [0, 17, 39, 58, 97, 98, 114, 131, 137] [4, 13, 40, 65, 92, 115, 117, 118, 144] [14, 41, 44, 66, 75, 87, 93, 116, 118] [42, 45, 60, 67, 75, 88, 101, 117, 138] [10, 43, 46, 61, 89, 101, 102, 118, 139] [2, 39, 44, 49, 95, 119, 121, 122, 129] [3, 45, 50, 73, 96, 120, 122, 123, 130] [22, 40, 46, 51, 77, 121, 123, 140, 149] [23, 41, 47, 52, 78, 106, 122, 141, 149] [1, 24, 42, 48, 106, 107, 115, 123, 142] [0, 20, 49, 53, 99, 124, 126, 127, 147] [21, 26, 50, 54, 76, 94, 100, 125, 127] [27, 51, 55, 65, 76, 95, 110, 126, 143] [2, 28, 52, 66, 96, 110, 111, 127, 144] [11, 24, 53, 71, 77, 97, 104, 128, 130] [25, 54, 56, 72, 77, 78, 98, 105, 129] [19, 55, 57, 68, 78, 99, 115, 130, 145] [1, 8, 56, 60, 108, 131, 133, 134, 139] [9, 35, 57, 61, 79, 82, 109, 132, 134] [36, 53, 58, 62, 79, 83, 119, 133, 146] [5, 37, 54, 59, 84, 119, 120, 134, 147] [3, 33, 60, 63, 80, 106, 113, 135, 137] [34, 40, 61, 64, 80, 81, 107, 114, 136] [7, 41, 62, 71, 81, 108, 124, 137, 148] [21, 38, 63, 74, 82, 83, 111, 118, 138] [31, 39, 64, 65, 75, 83, 84, 112, 139] [6, 17, 65, 68, 85, 90, 122, 140, 142] [18, 49, 66, 69, 85, 86, 91, 123, 141] [12, 50, 63, 67, 86, 92, 131, 142, 149] [9, 47, 68, 70, 87, 88, 120, 127, 143] [15, 48, 53, 69, 76, 88, 89, 121, 144] [34, 44, 52, 70, 77, 91, 92, 125, 145] [14, 29, 71, 73, 94, 95, 102, 134, 146] [22, 30, 60, 72, 79, 95, 96, 103, 147] [18, 26, 59, 73, 80, 98, 99, 132, 148] [25, 35, 43, 74, 85, 107, 108, 116, 149]
Code ID 150-8-18 · download JSON · raw on GitHub