← back to the board
[[180,16,14]] d ≤
n
180
k
16
d
14
kd²/n
17.422
w
7
X/Z
1

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 14, d_Z ≤ 14 · w_X = 7, w_Z = 7 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 14 · witness weight 14 (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, 14 qubits)
[7, 30, 31, 42, 50, 56, 57, 74, 86, 87, 101, 127, 142, 153]
d_Z 14 · witness weight 14 (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, 14 qubits)
[13, 26, 28, 35, 42, 45, 63, 67, 103, 125, 132, 153, 158, 170]
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 7 · H_Z 7
qubit degrees H_X 3–4 (mean 3.5) · H_Z 3–4 (mean 3.5)
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 (1,4): 90 (2,4): 360 (2,5): 1080 (2,6): 360 (3,3): 90 (3,4): 180 (3,5): 1890 (3,6): 9900 (3,7): 11430 (3,8): 2880 (3,9): 1080 (3,10): 90
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 (1,4): 90 (2,4): 360 (2,5): 1080 (2,6): 360 (3,3): 90 (3,4): 180 (3,5): 1890 (3,6): 9900 (3,7): 11430 (3,8): 2880 (3,9): 1080 (3,10): 90

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C30 x C3 = SmallGroup(90, 10) (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, 5, 27, 29}, b = {0, 23, 84} as element indices. Check weight 7. 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

[[180,16,14]] — two-block group-algebra code over C30 x C3

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/180-16-14.json.

Parity checks

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