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[[186,10,17]] d ≤
n
186
k
10
d
17
kd²/n
15.538
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 17, d_Z ≤ 17 · 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 17 · witness weight 17 (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, 17 qubits)
[4, 52, 53, 60, 63, 73, 80, 84, 87, 90, 92, 95, 100, 112, 141, 150, 174]
d_Z 17 · witness weight 17 (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, 17 qubits)
[7, 12, 21, 29, 34, 40, 44, 49, 54, 62, 71, 72, 76, 87, 119, 129, 171]
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): 93 (3,5): 1023 (3,6): 9579 (3,7): 14043 (3,8): 5022 (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): 93 (3,5): 1023 (3,6): 9579 (3,7): 14043 (3,8): 5022 (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, 4, 37}, b = {0, 2, 9, 82} 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

[[186,10,17]] — 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/186-10-17.json.

Parity checks

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