[[144,8,17]] · QEC Challenge

[[144,8,17]] d ≤

144

kd²/n

16.056

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 17, d_Z ≤ 17 · 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 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)

[15, 18, 22, 33, 35, 55, 60, 67, 69, 71, 76, 81, 86, 101, 113, 120, 126]

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)

[6, 8, 14, 27, 35, 36, 41, 81, 87, 93, 100, 103, 106, 121, 134, 136, 141]

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)×72 (2,4)×216 (3,3)×72 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 72 (1,6): 72 (2,4): 216 (2,5): 144 (2,7): 1008 (2,10): 1080 (3,3): 72 (3,4): 72 (3,5): 648 (3,6): 2016 (3,7): 360 (3,8): 9576 (3,9): 4032 (3,10): 720 (3,11): 28080 (3,12): 3720 (3,13): 1872 (3,14): 15840 (3,16): 1440

trapping sets H_Z (1,3)×72 (2,4)×216 (3,3)×72 (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 C12 x C6 = SmallGroup(72, 36) (order 72), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 144; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 3, 35, 55, 57, 61}, b = {0, 11, 23} 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

[[144,8,17]] — two-block group-algebra code over C12 x C6

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| = 144

Parameters are also spelled out in provenance.construction of codes/144-8-17.json.

Parity checks

X-checks 72 (max weight 9) · Z-checks 72 (max weight 9)

H_X (72 checks, sparse supports)

[0, 13, 17, 18, 41, 66, 72, 93, 101] [1, 21, 25, 28, 52, 70, 73, 104, 111] [2, 7, 26, 30, 54, 59, 74, 88, 108] [0, 3, 8, 34, 36, 39, 73, 75, 120] [4, 31, 35, 36, 37, 59, 74, 76, 113] [4, 5, 32, 38, 60, 71, 77, 114, 122] [6, 15, 36, 40, 63, 65, 78, 96, 119] [1, 7, 17, 44, 47, 50, 77, 79, 128] [8, 41, 45, 47, 49, 65, 78, 80, 124] [8, 9, 10, 31, 42, 62, 81, 85, 130] [2, 10, 19, 21, 24, 49, 78, 82, 105] [11, 21, 22, 46, 50, 66, 72, 83, 126] [11, 12, 23, 47, 67, 69, 84, 107, 127] [3, 6, 13, 22, 52, 56, 79, 85, 134] [4, 6, 14, 24, 53, 54, 80, 82, 86] [5, 14, 15, 25, 55, 58, 81, 87, 136] [5, 16, 21, 51, 54, 56, 83, 88, 131] [16, 17, 52, 55, 57, 69, 84, 89, 132] [6, 18, 30, 32, 35, 57, 84, 90, 115] [19, 32, 34, 54, 58, 70, 73, 91, 135] [3, 19, 20, 46, 51, 55, 92, 98, 117] [7, 12, 21, 34, 60, 64, 87, 93, 138] [8, 12, 22, 35, 61, 63, 89, 90, 94] [4, 9, 22, 23, 26, 29, 72, 95, 140] [9, 24, 32, 59, 63, 64, 91, 96, 137] [24, 25, 26, 27, 51, 60, 92, 97, 103] [1, 10, 26, 37, 43, 63, 90, 98, 123] [1, 11, 27, 39, 41, 64, 75, 91, 99] [12, 27, 28, 40, 42, 45, 92, 100, 125] [12, 29, 36, 41, 43, 62, 76, 101, 138] [29, 30, 42, 44, 63, 71, 77, 102, 139] [14, 18, 19, 31, 43, 65, 94, 98, 103] [15, 20, 31, 32, 44, 68, 95, 104, 142] [1, 15, 16, 19, 33, 66, 96, 99, 105] [17, 20, 33, 34, 45, 67, 97, 100, 106] [0, 35, 42, 65, 67, 68, 102, 107, 141] [5, 18, 36, 49, 53, 67, 100, 108, 131] [5, 19, 37, 50, 52, 68, 79, 102, 109] [11, 13, 16, 20, 37, 38, 74, 110, 133] [20, 39, 47, 52, 53, 66, 80, 111, 142] [13, 14, 39, 40, 62, 67, 81, 112, 118] [22, 28, 30, 41, 53, 69, 106, 108, 113] [2, 14, 23, 41, 42, 48, 75, 114, 143] [5, 23, 24, 30, 43, 70, 107, 109, 115] [2, 16, 25, 43, 44, 46, 76, 110, 116] [1, 13, 45, 46, 48, 69, 112, 117, 123] [7, 8, 27, 46, 56, 70, 85, 109, 118] [9, 28, 46, 47, 57, 61, 110, 119, 137] [6, 8, 28, 29, 48, 59, 86, 111, 120] [9, 30, 48, 49, 58, 60, 87, 112, 121] [2, 50, 55, 60, 61, 70, 89, 122, 143] [7, 32, 33, 37, 39, 51, 115, 118, 123] [34, 38, 40, 51, 52, 61, 116, 119, 124] [3, 9, 35, 40, 53, 71, 117, 121, 125] [15, 17, 37, 54, 64, 71, 93, 121, 126] [0, 27, 33, 38, 54, 55, 82, 127, 141] [12, 17, 38, 39, 56, 65, 94, 122, 128] [0, 29, 31, 40, 56, 57, 83, 95, 129] [6, 26, 31, 33, 58, 71, 97, 130, 134] [15, 42, 43, 49, 50, 59, 125, 126, 131] [10, 11, 33, 44, 59, 60, 86, 127, 132] [0, 7, 11, 45, 61, 62, 88, 129, 133] [18, 22, 24, 47, 48, 62, 103, 128, 134] [23, 25, 49, 62, 63, 68, 104, 129, 135] [10, 20, 25, 50, 64, 69, 106, 130, 136] [13, 23, 53, 57, 58, 65, 133, 135, 137] [28, 34, 35, 55, 56, 66, 113, 136, 138] [3, 4, 48, 57, 66, 67, 99, 114, 139] [2, 4, 18, 51, 58, 68, 101, 116, 140] [3, 21, 27, 29, 61, 69, 105, 139, 141] [26, 38, 44, 45, 64, 70, 124, 140, 142] [10, 14, 16, 36, 68, 71, 120, 132, 143]

H_Z (72 checks, sparse supports)

[0, 11, 23, 72, 75, 107, 127, 129, 133] [1, 3, 19, 73, 79, 98, 99, 105, 117] [2, 4, 38, 74, 82, 114, 116, 122, 140] [3, 27, 42, 75, 85, 92, 125, 139, 141] [4, 29, 44, 76, 77, 86, 95, 139, 140] [5, 7, 30, 77, 87, 88, 108, 109, 115] [6, 8, 10, 78, 85, 86, 90, 120, 130] [7, 13, 37, 74, 79, 93, 118, 123, 133] [8, 14, 39, 75, 80, 81, 94, 118, 120] [9, 15, 40, 81, 95, 96, 119, 121, 125] [10, 14, 55, 81, 82, 98, 132, 136, 143] [11, 16, 57, 83, 84, 99, 110, 132, 133] [12, 17, 18, 84, 93, 94, 100, 101, 128] [9, 13, 46, 72, 85, 110, 112, 117, 137] [14, 48, 60, 86, 87, 103, 112, 114, 143] [15, 21, 49, 78, 87, 104, 105, 126, 131] [2, 16, 61, 88, 89, 105, 110, 116, 143] [17, 22, 50, 72, 79, 89, 106, 126, 128] [18, 22, 26, 72, 90, 103, 108, 134, 140] [19, 24, 27, 82, 91, 92, 103, 105, 109] [20, 25, 28, 92, 104, 106, 110, 111, 136] [0, 21, 54, 73, 82, 83, 88, 93, 141] [22, 31, 56, 83, 85, 94, 95, 113, 134] [23, 32, 57, 84, 95, 114, 115, 135, 137] [6, 24, 33, 82, 86, 96, 97, 115, 134] [25, 34, 58, 73, 87, 97, 116, 135, 136] [20, 26, 31, 74, 95, 97, 98, 130, 142] [27, 33, 67, 97, 99, 100, 118, 127, 141] [28, 34, 36, 73, 100, 113, 119, 120, 138] [0, 29, 68, 95, 101, 102, 120, 129, 141] [30, 35, 37, 74, 90, 102, 113, 115, 121] [25, 31, 62, 76, 81, 103, 104, 129, 130] [1, 32, 63, 77, 90, 91, 96, 104, 123] [10, 33, 69, 105, 106, 123, 127, 130, 132] [34, 41, 64, 75, 91, 93, 106, 124, 138] [12, 35, 43, 76, 90, 94, 107, 125, 138] [2, 36, 41, 75, 76, 78, 101, 108, 143] [37, 43, 46, 76, 98, 109, 110, 123, 126] [38, 44, 47, 77, 110, 124, 127, 128, 142] [1, 39, 48, 75, 99, 111, 112, 123, 128] [40, 45, 49, 78, 100, 112, 124, 125, 129] [4, 41, 66, 72, 80, 99, 101, 113, 114] [5, 42, 67, 81, 100, 102, 107, 114, 131] [18, 43, 51, 98, 101, 103, 115, 116, 131] [44, 52, 68, 79, 102, 104, 116, 132, 142] [20, 45, 53, 80, 100, 106, 117, 133, 142] [40, 46, 51, 83, 92, 116, 117, 118, 119] [6, 47, 52, 79, 80, 84, 111, 119, 134] [3, 48, 71, 114, 117, 120, 121, 134, 139] [49, 53, 54, 80, 82, 108, 121, 131, 135] [5, 50, 56, 79, 83, 109, 122, 131, 136] [26, 45, 51, 88, 92, 97, 123, 124, 140] [8, 52, 70, 73, 85, 89, 109, 111, 124] [28, 53, 59, 86, 108, 111, 113, 125, 137] [11, 54, 59, 74, 86, 88, 91, 126, 127] [12, 55, 60, 87, 89, 92, 122, 127, 138] [7, 56, 62, 85, 88, 118, 128, 129, 138] [57, 61, 63, 89, 90, 119, 129, 137, 139] [9, 58, 64, 87, 91, 121, 130, 137, 140] [16, 36, 59, 74, 76, 96, 120, 131, 132] [17, 60, 71, 77, 93, 97, 121, 122, 132] [38, 61, 65, 94, 119, 122, 124, 133, 141] [13, 58, 62, 81, 101, 112, 133, 134, 135] [19, 63, 65, 78, 94, 96, 98, 102, 135] [15, 64, 66, 93, 96, 99, 126, 136, 142] [24, 47, 65, 78, 80, 103, 107, 128, 137] [21, 29, 66, 72, 83, 105, 111, 138, 139] [30, 67, 69, 84, 106, 107, 108, 112, 139] [23, 68, 70, 104, 107, 109, 135, 140, 143] [35, 55, 69, 84, 89, 113, 117, 136, 141] [32, 39, 70, 73, 91, 115, 118, 122, 142] [42, 50, 71, 77, 102, 125, 126, 130, 143]

Code ID 144-8-17 · download JSON · raw on GitHub