[[186,12,19]] · QEC Challenge

[[186,12,19]] d ≤

186

kd²/n

23.29

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 19, d_Z ≤ 19 · 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 19 · witness weight 19 (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, 19 qubits)

[14, 28, 62, 69, 70, 71, 73, 81, 82, 130, 131, 133, 136, 141, 142, 158, 159, 172, 180]

d_Z 19 · witness weight 19 (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, 19 qubits)

[29, 50, 51, 53, 55, 62, 63, 69, 83, 113, 120, 122, 127, 129, 135, 158, 172, 173, 180]

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)×186 (2,6)×2604 (3,6)×1860 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,4): 186 (2,6): 2604 (3,6): 1860 (3,8): 49104 (3,10): 5208

trapping sets H_Z (1,4)×186 (2,6)×2604 (3,6)×1860 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,4): 186 (2,6): 2604 (3,6): 1860 (3,8): 49104 (3,10): 5208

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, 7, 21}, b = {0, 6, 39, 54} 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 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

[[186,12,19]] — 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/186-12-19.json.

Parity checks

X-checks 93 (max weight 8) · Z-checks 93 (max weight 8)

H_X (93 checks, sparse supports)

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

H_Z (93 checks, sparse supports)

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

Code ID 186-12-19 · download JSON · raw on GitHub