[[186,30,11]] · QEC Challenge

[[186,30,11]] d ≤

186

kd²/n

19.516

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 11, d_Z ≤ 11 · w_X = 9, w_Z = 9 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 11 · witness weight 11 (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, 11 qubits)

[5, 16, 44, 50, 62, 68, 74, 86, 97, 109, 172]

d_Z 11 · witness weight 11 (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, 11 qubits)

[0, 11, 29, 93, 103, 115, 127, 133, 139, 151, 157]

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 9 · H_Z 9

qubit degrees H_X 3–6 (mean 4.5) · H_Z 3–6 (mean 4.5)

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 (1,6): 93 (2,4): 279 (2,7): 1674 (2,10): 1395 (3,3): 93 (3,5): 837 (3,6): 372 (3,7): 93 (3,8): 18135 (3,9): 1953 (3,10): 1674 (3,11): 46035 (3,12): 3658 (3,13): 4185 (3,14): 23901 (3,16): 1860

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

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, 10, 11, 29, 88}, b = {0, 23, 59} as element indices. Check weight 9. 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

[[186,30,11]] — two-block group-algebra code over C93

Direction & hypothesis

Cell weight-9plus 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/186-30-11.json.

Parity checks

X-checks 93 (max weight 9) · Z-checks 93 (max weight 9)

H_X (93 checks, sparse supports)

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

H_Z (93 checks, sparse supports)

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

Code ID 186-30-11 · download JSON · raw on GitHub