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[[180,4,15]] d ≤
n
180
k
4
d
15
kd²/n
5.0
w
5
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 15, d_Z ≤ 15 · w_X = 5, w_Z = 5 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 15 · witness weight 15 (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, 15 qubits)
[10, 12, 38, 40, 50, 60, 69, 71, 78, 84, 87, 97, 100, 109, 160]
d_Z 15 · witness weight 15 (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, 15 qubits)
[25, 32, 38, 79, 99, 119, 122, 126, 138, 143, 154, 166, 169, 170, 179]
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 8 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 5 · H_Z 5
qubit degrees H_X 2–3 (mean 2.5) · H_Z 2–3 (mean 2.5)
trapping sets H_X (1,2)×90 (2,2)×90 (3,2)×90 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 90 (1,3): 90 (2,2): 90 (2,3): 540 (2,4): 270 (3,2): 90 (3,3): 1620 (3,4): 2970 (3,5): 1350 (3,6): 540 (3,7): 90
trapping sets H_Z (1,2)×90 (2,2)×90 (3,2)×90 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 90 (1,3): 90 (2,2): 90 (2,3): 540 (2,4): 270 (3,2): 90 (3,3): 1620 (3,4): 2970 (3,5): 1350 (3,6): 540 (3,7): 90

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, 5, 37}, b = {0, 9} as element indices. Check weight 5. 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 ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[180,4,15]] — 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/180-4-15.json.

Parity checks

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