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[[168,12,18]] d ≤
n
168
k
12
d
18
kd²/n
23.143
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)
[0, 3, 18, 22, 23, 29, 37, 47, 51, 54, 67, 79, 113, 124, 140, 144, 148, 153]
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)
[9, 16, 32, 36, 43, 45, 91, 105, 109, 119, 127, 133, 134, 135, 138, 157, 164, 165]
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)×84 (2,5)×84 (3,5)×168 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 84 (1,5): 84 (2,5): 84 (2,6): 504 (2,7): 1512 (2,8): 840 (3,5): 168 (3,6): 420 (3,7): 2016 (3,8): 6720 (3,9): 23604 (3,10): 30576 (3,11): 11424 (3,12): 2688 (3,13): 840
trapping sets H_Z (1,4)×84 (2,5)×84 (3,5)×168 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 84 (1,5): 84 (2,5): 84 (2,6): 504 (2,7): 1512 (2,8): 840 (3,5): 168 (3,6): 420 (3,7): 2016 (3,8): 6720 (3,9): 23604 (3,10): 30576 (3,11): 11424 (3,12): 2688 (3,13): 840

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C42 x C2 = SmallGroup(84, 15) (order 84), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 168; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 14, 33, 64, 71}, b = {0, 13, 30, 69} 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

[[168,12,18]] — two-block group-algebra code over C42 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| = 168

Parameters are also spelled out in provenance.construction of codes/168-12-18.json.

Parity checks

X-checks 84 (max weight 9) · Z-checks 84 (max weight 9)
H_X (84 checks, sparse supports)
[0, 39, 49, 58, 73, 84, 106, 108, 161] [1, 33, 43, 64, 68, 85, 102, 112, 164] [2, 40, 48, 65, 67, 86, 99, 115, 154] [3, 17, 60, 69, 80, 87, 89, 117, 166] [4, 5, 51, 61, 70, 88, 92, 118, 120] [5, 34, 42, 59, 72, 89, 94, 121, 159] [6, 12, 54, 74, 77, 86, 90, 123, 167] [2, 7, 45, 55, 75, 91, 97, 114, 124] [8, 26, 51, 75, 76, 85, 92, 126, 162] [1, 9, 52, 60, 77, 87, 93, 111, 127] [10, 28, 38, 47, 83, 94, 96, 97, 152] [11, 13, 29, 71, 78, 95, 98, 102, 129] [3, 12, 14, 63, 72, 96, 103, 130, 132] [13, 20, 45, 70, 79, 84, 97, 132, 165] [0, 14, 46, 54, 80, 90, 98, 105, 133] [15, 22, 32, 53, 82, 92, 99, 101, 157] [8, 16, 23, 66, 81, 93, 100, 108, 135] [6, 9, 17, 57, 67, 101, 109, 126, 136] [18, 29, 37, 56, 81, 90, 102, 104, 143] [6, 19, 38, 63, 82, 91, 94, 103, 138] [7, 8, 20, 64, 71, 95, 104, 123, 139] [21, 24, 40, 50, 59, 86, 105, 107, 109] [10, 22, 25, 41, 79, 106, 110, 114, 141] [11, 23, 26, 74, 80, 107, 115, 142, 144] [23, 24, 31, 62, 78, 87, 108, 110, 149] [3, 25, 32, 57, 83, 88, 99, 109, 144] [4, 13, 26, 58, 66, 100, 110, 117, 145] [18, 27, 34, 44, 65, 89, 103, 111, 113] [15, 19, 28, 35, 76, 104, 112, 120, 147] [16, 20, 29, 69, 77, 113, 121, 138, 148] [15, 30, 41, 49, 68, 84, 100, 114, 116] [16, 18, 31, 50, 74, 101, 105, 115, 150] [17, 19, 32, 75, 79, 106, 116, 135, 151] [0, 33, 36, 52, 62, 93, 117, 119, 121] [21, 34, 37, 53, 83, 118, 122, 126, 153] [13, 22, 35, 38, 81, 119, 127, 154, 155] [10, 35, 36, 43, 73, 85, 95, 120, 122] [11, 24, 37, 44, 69, 96, 111, 121, 155] [12, 25, 38, 70, 76, 112, 122, 129, 156] [1, 30, 39, 46, 56, 98, 115, 123, 125] [27, 31, 40, 47, 82, 116, 124, 132, 158] [8, 28, 32, 41, 78, 125, 133, 150, 159] [2, 27, 42, 53, 61, 88, 112, 126, 128] [28, 30, 43, 62, 81, 113, 117, 127, 160] [6, 29, 31, 44, 83, 118, 128, 147, 161] [4, 45, 48, 64, 73, 104, 129, 131, 133] [24, 33, 46, 49, 65, 130, 134, 138, 162] [15, 25, 34, 47, 50, 87, 131, 139, 163] [5, 21, 47, 48, 55, 91, 106, 132, 134] [22, 36, 49, 56, 78, 107, 123, 133, 163] [3, 23, 37, 50, 82, 124, 134, 141, 164] [7, 42, 51, 58, 68, 110, 127, 135, 137] [18, 39, 43, 52, 59, 128, 136, 144, 165] [10, 19, 40, 44, 53, 90, 137, 145, 160] [9, 39, 54, 65, 72, 96, 124, 138, 140] [15, 40, 42, 55, 73, 125, 129, 139, 166] [16, 24, 41, 43, 56, 92, 130, 140, 158] [5, 12, 57, 60, 75, 116, 141, 143, 145] [1, 36, 45, 58, 61, 94, 142, 146, 150] [27, 37, 46, 59, 62, 95, 143, 151, 167] [14, 33, 59, 60, 67, 101, 118, 144, 146] [10, 34, 48, 61, 68, 119, 135, 145, 167] [11, 18, 35, 49, 62, 97, 136, 146, 153] [2, 17, 54, 63, 70, 122, 139, 147, 149] [0, 30, 51, 55, 64, 99, 140, 148, 155] [21, 31, 52, 56, 65, 100, 149, 156, 166] [1, 20, 51, 66, 80, 107, 136, 150, 152] [5, 27, 52, 54, 67, 102, 137, 141, 151] [28, 36, 53, 55, 68, 103, 142, 152, 165] [6, 14, 23, 69, 71, 84, 128, 153, 156] [7, 48, 57, 70, 72, 105, 154, 157, 160] [0, 26, 45, 71, 77, 113, 130, 155, 157] [2, 21, 46, 60, 72, 108, 131, 147, 156] [22, 30, 47, 61, 73, 109, 148, 157, 162] [3, 9, 29, 66, 74, 85, 134, 151, 158] [4, 42, 63, 67, 75, 111, 152, 159, 163] [7, 13, 32, 63, 76, 86, 119, 148, 160] [14, 39, 64, 66, 77, 114, 149, 153, 161] [16, 26, 35, 78, 79, 88, 140, 162, 164] [4, 8, 38, 57, 79, 89, 125, 142, 163] [9, 33, 58, 71, 80, 120, 143, 158, 164] [11, 20, 41, 76, 81, 91, 146, 161, 165] [17, 25, 44, 74, 82, 93, 131, 159, 166] [12, 19, 50, 69, 83, 98, 137, 154, 167]
H_Z (84 checks, sparse supports)
[0, 13, 30, 69, 84, 98, 117, 148, 155] [1, 8, 36, 74, 85, 93, 123, 142, 150] [2, 6, 21, 76, 86, 91, 126, 147, 156] [3, 9, 24, 47, 87, 96, 109, 134, 158] [4, 25, 42, 78, 88, 110, 129, 159, 163] [3, 5, 27, 79, 88, 89, 132, 141, 151] [6, 14, 18, 53, 90, 101, 103, 128, 153] [7, 19, 48, 81, 91, 104, 135, 154, 160] [4, 8, 15, 56, 92, 100, 104, 125, 163] [9, 16, 33, 82, 93, 101, 138, 158, 164] [5, 10, 19, 58, 94, 106, 120, 137, 145] [11, 20, 36, 59, 95, 107, 121, 146, 165] [10, 12, 37, 54, 90, 96, 122, 141, 167] [7, 10, 13, 62, 95, 97, 110, 119, 160] [11, 14, 39, 83, 96, 98, 144, 153, 161] [2, 15, 25, 64, 99, 112, 114, 131, 139] [16, 26, 30, 65, 100, 113, 115, 140, 162] [15, 17, 31, 60, 87, 101, 116, 147, 166] [1, 11, 18, 67, 102, 111, 115, 136, 146] [12, 19, 27, 68, 103, 112, 116, 137, 167] [18, 20, 28, 45, 97, 104, 113, 150, 165] [14, 21, 31, 70, 105, 118, 132, 149, 156] [0, 22, 32, 48, 99, 106, 119, 133, 157] [21, 23, 49, 66, 100, 107, 108, 134, 153] [0, 16, 24, 72, 105, 108, 121, 130, 140] [17, 21, 25, 73, 106, 109, 122, 131, 166] [22, 24, 26, 51, 92, 107, 110, 155, 162] [9, 27, 37, 75, 111, 124, 126, 143, 151] [1, 28, 38, 42, 94, 112, 125, 127, 152] [27, 29, 43, 71, 95, 102, 113, 128, 158] [7, 22, 30, 77, 114, 123, 127, 148, 157] [2, 23, 31, 39, 108, 115, 124, 128, 149] [30, 32, 40, 57, 99, 109, 116, 125, 160] [3, 26, 33, 43, 85, 117, 130, 144, 164] [4, 34, 44, 60, 89, 111, 118, 131, 145] [33, 35, 61, 76, 112, 119, 120, 146, 162] [4, 28, 36, 80, 117, 120, 133, 142, 152] [5, 29, 33, 37, 102, 118, 121, 134, 143] [34, 36, 38, 63, 94, 103, 119, 122, 163] [6, 20, 39, 49, 84, 123, 136, 138, 161] [7, 40, 50, 54, 86, 105, 124, 137, 139] [39, 41, 55, 79, 106, 114, 125, 140, 165] [8, 17, 34, 42, 89, 126, 135, 139, 159] [9, 35, 43, 51, 85, 120, 127, 136, 140] [42, 44, 52, 69, 111, 121, 128, 137, 166] [11, 38, 45, 55, 91, 97, 129, 142, 155] [12, 46, 56, 71, 98, 123, 130, 143, 156] [45, 47, 72, 82, 94, 124, 131, 132, 157] [12, 13, 40, 48, 86, 129, 132, 145, 154] [14, 41, 45, 49, 84, 114, 130, 133, 146] [46, 48, 50, 74, 105, 115, 131, 134, 167] [16, 32, 51, 61, 88, 92, 135, 148, 150] [17, 52, 62, 66, 93, 117, 136, 149, 151] [51, 53, 67, 83, 99, 118, 126, 137, 152] [19, 29, 46, 54, 90, 98, 138, 147, 151] [20, 47, 55, 63, 91, 132, 139, 148, 152] [54, 56, 64, 78, 102, 123, 133, 140, 149] [22, 50, 57, 67, 101, 109, 141, 154, 163] [23, 58, 68, 79, 84, 110, 135, 142, 164] [18, 57, 59, 80, 89, 105, 136, 143, 144] [23, 25, 52, 60, 87, 93, 141, 144, 156] [26, 53, 57, 61, 88, 126, 142, 145, 157] [58, 60, 62, 81, 108, 117, 127, 143, 146] [28, 44, 63, 72, 96, 103, 147, 159, 160] [29, 64, 73, 76, 85, 104, 129, 148, 161] [24, 63, 65, 77, 86, 111, 130, 138, 149] [31, 41, 58, 66, 100, 110, 150, 158, 161] [32, 59, 67, 74, 86, 101, 144, 151, 159] [10, 66, 68, 75, 85, 114, 135, 145, 152] [34, 62, 69, 77, 87, 113, 121, 153, 167] [2, 35, 70, 83, 88, 97, 122, 147, 154] [35, 37, 64, 71, 95, 104, 153, 155, 164] [38, 65, 69, 72, 89, 96, 138, 154, 156] [15, 70, 71, 73, 84, 120, 129, 139, 157] [40, 56, 74, 80, 90, 107, 115, 158, 166] [5, 41, 75, 82, 91, 92, 116, 141, 159] [43, 53, 70, 76, 92, 112, 122, 160, 165] [0, 44, 77, 81, 90, 93, 113, 155, 161] [8, 46, 73, 78, 95, 108, 125, 133, 162] [47, 49, 75, 79, 97, 106, 116, 162, 163] [1, 50, 78, 80, 87, 98, 107, 150, 164] [13, 52, 68, 81, 100, 102, 119, 127, 165] [3, 55, 65, 82, 99, 103, 124, 134, 166] [6, 59, 61, 83, 94, 109, 118, 128, 167]
Code ID 168-12-18 · download JSON · raw on GitHub