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[[180,22,14]] d ≤
n
180
k
22
d
14
kd²/n
23.956
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 14, d_Z ≤ 14 · 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 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)
[19, 53, 78, 80, 109, 114, 119, 122, 146, 149, 154, 158, 167, 170]
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)
[40, 41, 43, 44, 47, 50, 72, 73, 78, 88, 103, 114, 134, 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 6 · H_Z 6 (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 4 · H_Z 4
trapping sets H_X (1,4)×180 (2,6)×2520 (3,6)×1770 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 180 (2,6): 2520 (3,6): 1770 (3,8): 47610 (3,10): 5040
trapping sets H_Z (1,4)×180 (2,6)×2520 (3,6)×1770 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 180 (2,6): 2520 (3,6): 1770 (3,8): 47610 (3,10): 5040

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, 9, 12, 65}, b = {0, 3, 71, 83} 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

[[180,22,14]] — 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/180-22-14.json.

Parity checks

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