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[[180,12,18]] d ≤
n
180
k
12
d
18
kd²/n
21.6
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 18, d_Z ≤ 18 · w_X = 8, w_Z = 8 (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)
[11, 20, 21, 23, 24, 27, 34, 56, 58, 60, 61, 70, 71, 72, 124, 128, 134, 148]
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)
[7, 33, 38, 62, 63, 64, 93, 97, 104, 128, 133, 145, 147, 151, 152, 161, 174, 178]
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 8 · H_Z 8
qubit degrees H_X 3–5 (mean 4.0) · H_Z 3–5 (mean 4.0)
trapping sets H_X (1,3)×90 (2,4)×360 (3,3)×90 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 90 (1,5): 90 (2,4): 360 (2,6): 1170 (2,8): 900 (3,3): 90 (3,5): 1980 (3,7): 13005 (3,9): 24285 (3,11): 12510 (3,13): 900
trapping sets H_Z (1,3)×90 (2,4)×360 (3,3)×90 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 90 (1,5): 90 (2,4): 360 (2,6): 1170 (2,8): 900 (3,3): 90 (3,5): 1980 (3,7): 13005 (3,9): 24285 (3,11): 12510 (3,13): 900

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C15 x S3 = SmallGroup(90, 6) (order 90), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 180; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 10, 33, 36, 41}, b = {0, 23, 61} as element indices. Check weight 8. 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 2bga-nonabelian (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

[[180,12,18]] — two-block group-algebra code over C15 x S3

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| = 180

Parameters are also spelled out in provenance.construction of codes/180-12-18.json.

Parity checks

X-checks 90 (max weight 8) · Z-checks 90 (max weight 8)
H_X (90 checks, sparse supports)
[0, 17, 38, 39, 82, 90, 116, 178] [1, 11, 29, 48, 89, 91, 124, 179] [2, 13, 30, 54, 60, 92, 130, 165] [3, 21, 33, 56, 57, 93, 128, 133] [4, 8, 34, 58, 88, 94, 134, 160] [5, 7, 22, 62, 81, 95, 139, 171] [6, 25, 46, 47, 65, 96, 137, 142] [7, 26, 47, 66, 76, 97, 143, 166] [8, 24, 25, 45, 73, 98, 145, 174] [4, 9, 27, 48, 70, 99, 103, 148] [0, 10, 49, 71, 75, 100, 132, 149] [2, 11, 37, 51, 72, 101, 146, 150] [12, 20, 38, 52, 73, 98, 102, 151] [13, 21, 53, 70, 74, 101, 103, 172] [14, 16, 33, 36, 87, 104, 153, 177] [15, 18, 19, 39, 76, 105, 109, 156] [16, 19, 40, 51, 77, 106, 141, 157] [5, 17, 42, 63, 64, 107, 154, 158] [14, 18, 43, 64, 79, 104, 108, 159] [14, 19, 44, 80, 85, 107, 109, 175] [10, 20, 41, 42, 62, 110, 114, 161] [2, 21, 43, 63, 84, 111, 147, 162] [8, 12, 22, 44, 65, 112, 117, 163] [3, 13, 23, 66, 82, 90, 113, 164] [4, 24, 42, 67, 83, 112, 114, 150] [1, 9, 25, 55, 83, 94, 115, 162] [10, 26, 36, 56, 68, 110, 116, 165] [8, 27, 37, 69, 84, 111, 115, 117] [28, 31, 32, 51, 54, 118, 122, 167] [29, 32, 52, 55, 65, 119, 155, 168] [14, 30, 34, 35, 57, 120, 125, 169] [1, 31, 35, 58, 85, 91, 121, 170] [1, 32, 59, 68, 86, 120, 122, 158] [0, 15, 33, 77, 78, 97, 123, 168] [16, 28, 34, 60, 78, 118, 124, 171] [28, 29, 35, 61, 87, 119, 123, 125] [0, 23, 36, 59, 76, 126, 131, 172] [9, 24, 37, 60, 77, 92, 127, 173] [10, 38, 57, 61, 78, 126, 128, 163] [5, 20, 26, 39, 61, 100, 129, 134] [11, 21, 27, 40, 79, 93, 130, 174] [0, 12, 41, 80, 88, 94, 129, 131] [6, 22, 42, 71, 88, 102, 132, 173] [7, 23, 43, 54, 72, 103, 126, 133] [20, 24, 44, 55, 81, 127, 132, 134] [1, 45, 49, 50, 70, 135, 140, 175] [5, 46, 50, 68, 71, 95, 136, 176] [5, 47, 69, 72, 79, 135, 137, 169] [2, 28, 48, 52, 53, 106, 138, 143] [6, 29, 49, 53, 73, 96, 139, 177] [6, 7, 50, 74, 89, 97, 138, 140] [3, 30, 51, 85, 86, 108, 141, 176] [4, 31, 45, 52, 86, 109, 135, 142] [32, 45, 46, 53, 75, 136, 141, 143] [3, 14, 40, 54, 74, 111, 144, 149] [4, 22, 41, 55, 85, 99, 145, 178] [2, 23, 56, 75, 86, 100, 144, 146] [15, 36, 43, 57, 75, 113, 147, 151] [16, 25, 37, 44, 58, 101, 114, 148] [3, 26, 38, 59, 87, 102, 147, 149] [18, 40, 60, 70, 83, 117, 144, 150] [19, 36, 41, 61, 71, 90, 145, 151] [6, 8, 62, 66, 67, 119, 152, 157] [7, 15, 63, 67, 82, 105, 153, 179] [15, 16, 64, 81, 83, 106, 152, 154] [9, 45, 65, 68, 69, 121, 155, 159] [10, 17, 46, 66, 69, 107, 122, 156] [17, 18, 47, 67, 84, 108, 155, 157] [12, 49, 62, 68, 89, 125, 152, 158] [13, 50, 62, 63, 69, 91, 153, 159] [11, 28, 58, 70, 84, 127, 160, 164] [12, 29, 39, 59, 71, 112, 128, 161] [9, 13, 40, 72, 89, 113, 160, 162] [31, 42, 55, 61, 73, 115, 131, 163] [11, 32, 43, 56, 74, 92, 116, 164] [35, 54, 59, 75, 82, 93, 161, 165] [17, 20, 76, 79, 80, 136, 166, 170] [18, 21, 30, 77, 80, 120, 137, 167] [19, 30, 31, 78, 88, 121, 166, 168] [23, 33, 63, 79, 81, 123, 140, 169] [24, 33, 34, 64, 80, 95, 124, 170] [27, 67, 76, 77, 81, 96, 167, 171] [26, 46, 57, 74, 82, 129, 146, 172] [22, 27, 47, 58, 83, 98, 130, 173] [25, 50, 60, 72, 84, 99, 133, 174] [34, 37, 48, 85, 87, 138, 154, 175] [35, 38, 48, 49, 86, 104, 139, 176] [41, 51, 52, 78, 87, 105, 142, 177] [39, 44, 64, 73, 88, 110, 148, 178] [53, 56, 65, 66, 89, 118, 156, 179]
H_Z (90 checks, sparse supports)
[0, 23, 61, 90, 100, 123, 126, 131] [1, 31, 69, 91, 115, 121, 122, 135] [2, 37, 74, 92, 101, 111, 138, 146] [3, 40, 75, 93, 113, 141, 144, 149] [4, 25, 41, 94, 99, 114, 142, 145] [5, 46, 80, 95, 107, 129, 136, 137] [6, 49, 81, 96, 132, 139, 140, 152] [7, 33, 50, 95, 97, 133, 140, 153] [8, 12, 83, 94, 98, 112, 117, 152] [9, 55, 84, 99, 115, 127, 155, 162] [10, 39, 56, 100, 110, 116, 128, 156] [11, 13, 58, 91, 101, 130, 160, 164] [12, 42, 59, 102, 112, 131, 158, 161] [9, 13, 43, 92, 103, 113, 159, 162] [14, 18, 86, 104, 108, 109, 120, 144] [15, 63, 87, 105, 123, 147, 153, 154] [16, 48, 64, 104, 106, 124, 148, 154] [17, 19, 66, 90, 107, 156, 157, 166] [18, 51, 67, 105, 108, 150, 157, 167] [15, 19, 52, 105, 106, 109, 151, 168] [20, 26, 88, 102, 110, 129, 134, 166] [21, 27, 54, 93, 103, 111, 130, 167] [22, 24, 71, 95, 112, 132, 145, 173] [23, 57, 72, 113, 126, 133, 146, 169] [20, 24, 58, 98, 114, 127, 134, 170] [25, 27, 73, 96, 98, 115, 148, 174] [0, 26, 74, 97, 116, 129, 149, 172] [22, 27, 60, 99, 117, 130, 171, 173] [28, 34, 89, 118, 124, 125, 138, 160] [29, 35, 62, 91, 119, 125, 139, 161] [30, 32, 77, 92, 120, 141, 167, 168] [31, 65, 78, 118, 121, 142, 163, 168] [28, 32, 66, 118, 119, 122, 143, 164] [33, 35, 79, 93, 104, 123, 169, 170] [1, 34, 80, 94, 120, 124, 170, 175] [30, 35, 68, 120, 121, 125, 165, 176] [36, 38, 43, 104, 116, 126, 147, 151] [37, 44, 70, 101, 117, 127, 148, 175] [3, 38, 71, 90, 102, 128, 149, 176] [39, 41, 82, 90, 105, 129, 161, 178] [2, 40, 83, 106, 130, 144, 150, 162] [36, 41, 73, 110, 131, 145, 151, 177] [10, 42, 44, 107, 110, 114, 132, 163] [3, 43, 84, 108, 111, 133, 147, 164] [4, 39, 44, 109, 112, 134, 148, 178] [45, 47, 52, 98, 135, 142, 143, 155] [46, 53, 76, 96, 136, 143, 156, 172] [6, 47, 77, 96, 97, 137, 157, 173] [48, 50, 85, 91, 99, 138, 175, 176] [5, 49, 86, 100, 135, 139, 158, 176] [45, 50, 79, 135, 136, 140, 159, 174] [16, 51, 53, 101, 106, 118, 141, 177] [6, 52, 87, 102, 119, 138, 142, 177] [7, 48, 53, 103, 138, 139, 143, 179] [54, 56, 60, 92, 118, 133, 144, 165] [8, 55, 61, 115, 119, 134, 145, 163] [11, 56, 82, 93, 116, 146, 164, 179] [21, 57, 59, 93, 120, 128, 147, 172] [9, 58, 88, 94, 121, 148, 160, 173] [10, 54, 59, 122, 126, 149, 161, 165] [11, 24, 60, 92, 124, 127, 150, 174] [12, 57, 61, 125, 128, 129, 151, 163] [62, 64, 68, 95, 110, 152, 158, 159] [14, 63, 69, 107, 111, 153, 159, 169] [17, 64, 85, 107, 108, 154, 170, 178] [29, 65, 67, 96, 112, 119, 155, 179] [15, 66, 89, 97, 113, 152, 156, 179] [16, 62, 67, 114, 152, 153, 157, 171] [17, 32, 68, 116, 122, 136, 155, 158] [18, 65, 69, 117, 137, 155, 156, 159] [4, 70, 72, 99, 103, 135, 150, 160] [20, 71, 75, 100, 132, 136, 151, 161] [21, 25, 72, 101, 133, 137, 162, 174] [22, 38, 73, 98, 102, 139, 163, 178] [23, 70, 74, 103, 140, 144, 164, 172] [2, 26, 75, 100, 143, 146, 147, 165] [7, 76, 78, 97, 105, 126, 166, 171] [28, 77, 81, 106, 123, 127, 167, 171] [29, 33, 78, 123, 124, 128, 168, 177] [30, 47, 79, 108, 130, 137, 166, 169] [31, 76, 80, 109, 131, 166, 167, 170] [5, 34, 81, 95, 134, 154, 169, 171] [13, 36, 82, 90, 113, 153, 165, 172] [37, 42, 83, 114, 115, 150, 154, 173] [8, 40, 84, 111, 117, 157, 160, 174] [19, 45, 85, 109, 121, 141, 145, 175] [46, 51, 86, 122, 141, 142, 146, 176] [14, 49, 87, 104, 125, 149, 175, 177] [0, 55, 88, 94, 131, 132, 168, 178] [1, 63, 89, 91, 140, 158, 162, 179]
Code ID 180-12-18 · download JSON · raw on GitHub