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[[186,20,12]] d ≤
n
186
k
20
d
12
kd²/n
15.484
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 12, d_Z ≤ 12 · 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 12 · witness weight 12 (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, 12 qubits)
[29, 30, 35, 49, 50, 62, 70, 76, 117, 123, 137, 157]
d_Z 12 · witness weight 12 (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, 12 qubits)
[32, 36, 56, 60, 77, 81, 121, 123, 142, 144, 145, 147]
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 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)×93 (2,4)×279 (3,3)×93 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 93 (1,4): 93 (2,4): 279 (2,5): 1116 (2,6): 558 (3,3): 93 (3,4): 186 (3,5): 837 (3,6): 9331 (3,7): 14601 (3,8): 4929 (3,9): 1674 (3,10): 372
trapping sets H_Z (1,3)×93 (2,4)×279 (3,3)×93 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 93 (1,4): 93 (2,4): 279 (2,5): 1116 (2,6): 558 (3,3): 93 (3,4): 186 (3,5): 837 (3,6): 9331 (3,7): 14601 (3,8): 4929 (3,9): 1674 (3,10): 372

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C93 = SmallGroup(93, 2) (order 93), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 186; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 1, 21, 66}, b = {0, 5, 26} 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

[[186,20,12]] — two-block group-algebra code over C93

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| = 186

Parameters are also spelled out in provenance.construction of codes/186-20-12.json.

Parity checks

X-checks 93 (max weight 7) · Z-checks 93 (max weight 7)
H_X (93 checks, sparse supports)
[0, 3, 31, 76, 93, 158, 179] [0, 1, 33, 78, 94, 160, 181] [2, 6, 34, 79, 95, 161, 182] [1, 3, 29, 74, 96, 162, 183] [2, 4, 36, 81, 97, 163, 184] [5, 9, 37, 82, 93, 98, 164] [4, 6, 32, 77, 99, 165, 185] [5, 7, 39, 84, 94, 100, 166] [8, 12, 40, 85, 95, 101, 167] [7, 9, 35, 80, 96, 102, 168] [8, 10, 42, 87, 97, 103, 169] [11, 15, 43, 88, 98, 104, 170] [10, 12, 38, 83, 99, 105, 171] [11, 13, 45, 90, 100, 106, 172] [14, 18, 46, 91, 101, 107, 173] [13, 15, 41, 86, 102, 108, 174] [14, 16, 48, 92, 103, 109, 175] [1, 17, 21, 49, 104, 110, 176] [16, 18, 44, 89, 105, 111, 177] [3, 17, 19, 51, 106, 112, 178] [4, 20, 24, 52, 107, 113, 179] [0, 19, 21, 47, 108, 114, 180] [6, 20, 22, 54, 109, 115, 181] [7, 23, 27, 55, 110, 116, 182] [2, 22, 24, 50, 111, 117, 183] [9, 23, 25, 57, 112, 118, 184] [10, 26, 30, 58, 93, 113, 119] [5, 25, 27, 53, 114, 120, 185] [12, 26, 28, 60, 94, 115, 121] [13, 29, 33, 61, 95, 116, 122] [8, 28, 30, 56, 96, 117, 123] [15, 29, 31, 63, 97, 118, 124] [16, 32, 36, 64, 98, 119, 125] [11, 31, 33, 59, 99, 120, 126] [18, 32, 34, 66, 100, 121, 127] [19, 35, 39, 67, 101, 122, 128] [14, 34, 36, 62, 102, 123, 129] [21, 35, 37, 69, 103, 124, 130] [22, 38, 42, 70, 104, 125, 131] [17, 37, 39, 65, 105, 126, 132] [24, 38, 40, 72, 106, 127, 133] [25, 41, 45, 73, 107, 128, 134] [20, 40, 42, 68, 108, 129, 135] [27, 41, 43, 75, 109, 130, 136] [28, 44, 48, 76, 110, 131, 137] [23, 43, 45, 71, 111, 132, 138] [30, 44, 46, 78, 112, 133, 139] [31, 47, 51, 79, 113, 134, 140] [26, 46, 48, 74, 114, 135, 141] [33, 47, 49, 81, 115, 136, 142] [34, 50, 54, 82, 116, 137, 143] [29, 49, 51, 77, 117, 138, 144] [36, 50, 52, 84, 118, 139, 145] [37, 53, 57, 85, 119, 140, 146] [32, 52, 54, 80, 120, 141, 147] [39, 53, 55, 87, 121, 142, 148] [40, 56, 60, 88, 122, 143, 149] [35, 55, 57, 83, 123, 144, 150] [42, 56, 58, 90, 124, 145, 151] [43, 59, 63, 91, 125, 146, 152] [38, 58, 60, 86, 126, 147, 153] [45, 59, 61, 92, 127, 148, 154] [1, 46, 62, 66, 128, 149, 155] [41, 61, 63, 89, 129, 150, 156] [3, 48, 62, 64, 130, 151, 157] [4, 49, 65, 69, 131, 152, 158] [0, 44, 64, 66, 132, 153, 159] [6, 51, 65, 67, 133, 154, 160] [7, 52, 68, 72, 134, 155, 161] [2, 47, 67, 69, 135, 156, 162] [9, 54, 68, 70, 136, 157, 163] [10, 55, 71, 75, 137, 158, 164] [5, 50, 70, 72, 138, 159, 165] [12, 57, 71, 73, 139, 160, 166] [13, 58, 74, 78, 140, 161, 167] [8, 53, 73, 75, 141, 162, 168] [15, 60, 74, 76, 142, 163, 169] [16, 61, 77, 81, 143, 164, 170] [11, 56, 76, 78, 144, 165, 171] [18, 63, 77, 79, 145, 166, 172] [19, 64, 80, 84, 146, 167, 173] [14, 59, 79, 81, 147, 168, 174] [21, 66, 80, 82, 148, 169, 175] [22, 67, 83, 87, 149, 170, 176] [17, 62, 82, 84, 150, 171, 177] [24, 69, 83, 85, 151, 172, 178] [25, 70, 86, 90, 152, 173, 179] [20, 65, 85, 87, 153, 174, 180] [27, 72, 86, 88, 154, 175, 181] [28, 73, 89, 92, 155, 176, 182] [23, 68, 88, 90, 156, 177, 183] [30, 75, 89, 91, 157, 178, 184] [26, 71, 91, 92, 159, 180, 185]
H_Z (93 checks, sparse supports)
[0, 5, 26, 93, 94, 114, 159] [1, 7, 28, 94, 96, 110, 155] [2, 8, 29, 95, 97, 117, 162] [3, 9, 30, 93, 96, 112, 157] [4, 10, 31, 97, 99, 113, 158] [5, 11, 32, 98, 100, 120, 165] [6, 12, 33, 95, 99, 115, 160] [7, 13, 34, 100, 102, 116, 161] [8, 14, 35, 101, 103, 123, 168] [9, 15, 36, 98, 102, 118, 163] [10, 16, 37, 103, 105, 119, 164] [11, 17, 38, 104, 106, 126, 171] [12, 18, 39, 101, 105, 121, 166] [13, 19, 40, 106, 108, 122, 167] [14, 20, 41, 107, 109, 129, 174] [15, 21, 42, 104, 108, 124, 169] [16, 22, 43, 109, 111, 125, 170] [17, 23, 44, 110, 112, 132, 177] [18, 24, 45, 107, 111, 127, 172] [19, 25, 46, 112, 114, 128, 173] [20, 26, 47, 113, 115, 135, 180] [21, 27, 48, 110, 114, 130, 175] [22, 28, 49, 115, 117, 131, 176] [23, 29, 50, 116, 118, 138, 183] [24, 30, 51, 113, 117, 133, 178] [25, 31, 52, 118, 120, 134, 179] [26, 32, 53, 119, 121, 141, 185] [27, 33, 54, 116, 120, 136, 181] [28, 34, 55, 121, 123, 137, 182] [29, 35, 56, 96, 122, 124, 144] [30, 36, 57, 119, 123, 139, 184] [31, 37, 58, 93, 124, 126, 140] [32, 38, 59, 99, 125, 127, 147] [33, 39, 60, 94, 122, 126, 142] [34, 40, 61, 95, 127, 129, 143] [35, 41, 62, 102, 128, 130, 150] [36, 42, 63, 97, 125, 129, 145] [37, 43, 64, 98, 130, 132, 146] [38, 44, 65, 105, 131, 133, 153] [39, 45, 66, 100, 128, 132, 148] [40, 46, 67, 101, 133, 135, 149] [41, 47, 68, 108, 134, 136, 156] [42, 48, 69, 103, 131, 135, 151] [43, 49, 70, 104, 136, 138, 152] [44, 50, 71, 111, 137, 139, 159] [45, 51, 72, 106, 134, 138, 154] [46, 52, 73, 107, 139, 141, 155] [47, 53, 74, 114, 140, 142, 162] [48, 54, 75, 109, 137, 141, 157] [49, 55, 76, 110, 142, 144, 158] [50, 56, 77, 117, 143, 145, 165] [51, 57, 78, 112, 140, 144, 160] [52, 58, 79, 113, 145, 147, 161] [53, 59, 80, 120, 146, 148, 168] [54, 60, 81, 115, 143, 147, 163] [55, 61, 82, 116, 148, 150, 164] [56, 62, 83, 123, 149, 151, 171] [57, 63, 84, 118, 146, 150, 166] [58, 64, 85, 119, 151, 153, 167] [59, 65, 86, 126, 152, 154, 174] [60, 66, 87, 121, 149, 153, 169] [61, 67, 88, 122, 154, 156, 170] [62, 68, 89, 129, 155, 157, 177] [63, 69, 90, 124, 152, 156, 172] [64, 70, 91, 125, 157, 159, 173] [0, 65, 71, 132, 158, 160, 180] [66, 72, 92, 127, 155, 159, 175] [1, 67, 73, 128, 160, 162, 176] [2, 68, 74, 135, 161, 163, 183] [3, 69, 75, 130, 158, 162, 178] [4, 70, 76, 131, 163, 165, 179] [5, 71, 77, 138, 164, 166, 185] [6, 72, 78, 133, 161, 165, 181] [7, 73, 79, 134, 166, 168, 182] [8, 74, 80, 96, 141, 167, 169] [9, 75, 81, 136, 164, 168, 184] [10, 76, 82, 93, 137, 169, 171] [11, 77, 83, 99, 144, 170, 172] [12, 78, 84, 94, 139, 167, 171] [13, 79, 85, 95, 140, 172, 174] [14, 80, 86, 102, 147, 173, 175] [15, 81, 87, 97, 142, 170, 174] [16, 82, 88, 98, 143, 175, 177] [17, 83, 89, 105, 150, 176, 178] [18, 84, 90, 100, 145, 173, 177] [19, 85, 91, 101, 146, 178, 180] [0, 20, 86, 108, 153, 179, 181] [21, 87, 92, 103, 148, 176, 180] [1, 22, 88, 104, 149, 181, 183] [2, 23, 89, 111, 156, 182, 184] [3, 24, 90, 106, 151, 179, 183] [4, 25, 91, 107, 152, 184, 185] [6, 27, 92, 109, 154, 182, 185]
Code ID 186-20-12 · download JSON · raw on GitHub