[[112,18,11]] · QEC Challenge

[[112,18,11]] d ≤

112

kd²/n

19.446

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, 11, 17, 23, 31, 51, 61, 72, 99, 102, 110]

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)

[4, 21, 31, 34, 44, 69, 71, 72, 89, 107, 111]

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 4 · H_Z 4 (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)×56 (2,4)×168 (3,3)×56 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 56 (1,6): 56 (2,4): 168 (2,5): 56 (2,7): 896 (2,8): 140 (2,10): 560 (3,3): 56 (3,5): 504 (3,6): 1008 (3,7): 168 (3,8): 9240 (3,9): 5824 (3,10): 1512 (3,11): 15792 (3,12): 3696 (3,13): 1400 (3,14): 5936 (3,16): 336

trapping sets H_Z (1,3)×56 (2,4)×168 (3,3)×56 (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 C14 x C2 x C2 = SmallGroup(56, 13) (order 56), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 112; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 8, 13, 23, 34, 48}, b = {0, 7, 25} 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

[[112,18,11]] — two-block group-algebra code over C14 x C2 x C2

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| = 112

Parameters are also spelled out in provenance.construction of codes/112-18-11.json.

Parity checks

X-checks 56 (max weight 9) · Z-checks 56 (max weight 9)

H_X (56 checks, sparse supports)

[0, 8, 24, 26, 31, 52, 56, 89, 103] [1, 12, 27, 28, 30, 49, 57, 93, 99] [2, 3, 18, 32, 37, 47, 58, 83, 108] [2, 3, 17, 19, 38, 55, 59, 96, 109] [4, 5, 16, 32, 34, 39, 57, 60, 97] [5, 6, 22, 33, 36, 43, 61, 87, 105] [5, 6, 21, 23, 34, 54, 62, 100, 106] [2, 7, 20, 35, 36, 38, 56, 63, 101] [0, 8, 11, 25, 44, 53, 64, 90, 111] [1, 9, 10, 26, 40, 45, 61, 65, 91] [9, 10, 12, 25, 27, 46, 62, 66, 104] [11, 13, 24, 40, 42, 47, 63, 67, 105] [1, 12, 15, 29, 40, 50, 68, 94, 110] [0, 13, 14, 30, 41, 44, 58, 69, 95] [8, 13, 14, 29, 31, 42, 59, 70, 107] [9, 15, 28, 43, 44, 46, 60, 71, 108] [4, 6, 16, 19, 33, 51, 68, 72, 98] [7, 17, 18, 34, 48, 52, 69, 73, 99] [17, 18, 20, 33, 35, 53, 70, 74, 110] [1, 19, 21, 32, 48, 50, 58, 71, 75] [3, 7, 20, 23, 37, 48, 64, 76, 102] [4, 21, 22, 38, 49, 51, 65, 77, 103] [16, 21, 22, 37, 39, 50, 66, 78, 111] [0, 17, 23, 36, 51, 53, 61, 67, 79] [11, 14, 24, 27, 41, 55, 76, 80, 106] [5, 15, 25, 26, 42, 54, 56, 77, 81] [6, 25, 26, 28, 41, 43, 64, 78, 82] [3, 7, 27, 29, 40, 54, 65, 79, 83] [10, 15, 28, 31, 45, 54, 72, 84, 109] [2, 11, 29, 30, 46, 55, 57, 73, 85] [3, 24, 29, 30, 45, 47, 68, 74, 86] [4, 6, 25, 31, 44, 55, 69, 75, 87] [12, 19, 22, 32, 35, 49, 59, 84, 88] [8, 13, 23, 33, 34, 50, 60, 85, 89] [0, 14, 33, 34, 36, 49, 72, 86, 90] [8, 10, 15, 35, 37, 48, 73, 87, 91] [8, 18, 23, 36, 39, 52, 62, 80, 92] [9, 12, 19, 37, 38, 53, 63, 81, 93] [1, 10, 32, 37, 38, 52, 76, 82, 94] [11, 12, 14, 33, 39, 51, 77, 83, 95] [2, 20, 27, 30, 40, 43, 66, 92, 96] [3, 16, 21, 31, 41, 42, 67, 93, 97] [2, 4, 22, 41, 42, 44, 80, 94, 98] [16, 18, 23, 43, 45, 54, 81, 95, 99] [5, 16, 26, 31, 44, 47, 70, 88, 100] [6, 17, 20, 27, 45, 46, 71, 89, 101] [5, 7, 18, 40, 45, 46, 84, 90, 102] [19, 20, 22, 41, 47, 55, 85, 91, 103] [0, 9, 28, 35, 38, 48, 74, 100, 104] [10, 24, 29, 39, 49, 50, 75, 101, 105] [9, 11, 30, 49, 50, 51, 88, 102, 106] [1, 13, 24, 34, 39, 51, 78, 96, 107] [14, 25, 28, 35, 52, 53, 79, 97, 108] [13, 15, 26, 48, 52, 53, 92, 98, 109] [4, 17, 36, 43, 46, 54, 82, 107, 110] [7, 21, 32, 42, 47, 55, 86, 104, 111]

H_Z (56 checks, sparse supports)

[0, 7, 25, 56, 64, 69, 79, 90, 104] [1, 4, 29, 57, 65, 68, 75, 94, 107] [2, 13, 19, 58, 59, 63, 85, 96, 98] [3, 14, 32, 58, 59, 76, 83, 86, 97] [4, 15, 33, 60, 72, 77, 87, 98, 110] [5, 9, 23, 60, 61, 62, 81, 100, 102] [6, 10, 36, 61, 62, 72, 82, 87, 101] [7, 11, 37, 63, 73, 76, 83, 102, 111] [8, 20, 26, 56, 64, 70, 89, 91, 92] [9, 21, 27, 65, 66, 71, 93, 104, 106] [10, 22, 40, 65, 66, 84, 91, 94, 105] [11, 23, 41, 64, 67, 80, 85, 95, 106] [12, 16, 30, 57, 66, 68, 88, 93, 95] [13, 17, 31, 67, 69, 70, 89, 107, 109] [14, 18, 44, 69, 70, 80, 90, 95, 108] [15, 19, 45, 68, 71, 81, 84, 91, 109] [16, 28, 34, 60, 72, 78, 97, 99, 100] [17, 29, 35, 59, 73, 74, 79, 101, 110] [18, 30, 48, 58, 73, 74, 92, 99, 102] [19, 31, 49, 59, 72, 75, 88, 93, 103] [20, 24, 38, 63, 74, 76, 96, 101, 103] [21, 25, 39, 62, 75, 77, 78, 97, 111] [22, 26, 51, 61, 77, 78, 88, 98, 103] [23, 27, 52, 62, 76, 79, 89, 92, 99] [24, 36, 42, 56, 67, 80, 86, 105, 107] [25, 37, 43, 64, 66, 81, 82, 87, 108] [26, 38, 54, 56, 65, 81, 82, 100, 109] [2, 27, 39, 57, 66, 80, 83, 96, 101] [28, 32, 46, 57, 71, 82, 84, 104, 108] [29, 33, 47, 68, 70, 83, 85, 86, 105] [30, 34, 55, 57, 69, 85, 86, 96, 106] [5, 31, 35, 56, 70, 84, 87, 97, 100] [32, 44, 50, 58, 60, 75, 88, 94, 111] [0, 33, 45, 61, 72, 74, 89, 90, 95] [8, 34, 46, 60, 62, 73, 89, 90, 107] [9, 35, 47, 63, 74, 88, 91, 104, 108] [36, 40, 53, 61, 63, 79, 90, 92, 110] [1, 37, 41, 58, 76, 78, 91, 93, 94] [12, 38, 42, 59, 63, 77, 93, 94, 104] [13, 39, 43, 60, 78, 92, 95, 105, 107] [3, 40, 51, 65, 67, 68, 83, 96, 102] [4, 41, 52, 69, 80, 82, 97, 98, 103] [16, 42, 53, 67, 70, 81, 97, 98, 111] [1, 17, 43, 61, 71, 82, 96, 99, 110] [6, 44, 48, 64, 69, 71, 87, 98, 100] [7, 45, 49, 65, 84, 86, 99, 101, 102] [20, 46, 50, 66, 71, 85, 101, 102, 110] [0, 21, 47, 58, 67, 86, 100, 103, 111] [10, 48, 55, 73, 75, 76, 91, 104, 109] [5, 11, 49, 57, 77, 88, 90, 105, 106] [6, 24, 50, 68, 75, 78, 89, 105, 106] [14, 51, 54, 72, 77, 79, 95, 106, 107] [2, 15, 52, 56, 73, 92, 94, 108, 109] [3, 28, 53, 64, 74, 79, 93, 108, 109] [12, 18, 54, 62, 81, 83, 84, 99, 110] [8, 22, 55, 59, 80, 85, 87, 103, 111]

Code ID 112-18-11 · download JSON · raw on GitHub