[[168,24,8]] · QEC Challenge

[[168,24,8]] d ≤

168

kd²/n

9.143

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 8, d_Z ≤ 8 · 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 8 · witness weight 8 (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, 8 qubits)

[89, 97, 98, 109, 122, 133, 157, 164]

d_Z 8 · witness weight 8 (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, 8 qubits)

[3, 10, 22, 33, 34, 45, 46, 57]

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)×84 (2,4)×252 (3,3)×84 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 84 (1,4): 84 (2,4): 252 (2,5): 1008 (2,6): 504 (3,3): 84 (3,4): 84 (3,5): 756 (3,6): 9016 (3,7): 13188 (3,8): 3444 (3,9): 1512 (3,10): 336

trapping sets H_Z (1,3)×84 (2,4)×252 (3,3)×84 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 84 (1,4): 84 (2,4): 252 (2,5): 1008 (2,6): 504 (3,3): 84 (3,4): 84 (3,5): 756 (3,6): 9016 (3,7): 13188 (3,8): 3444 (3,9): 1512 (3,10): 336

Construction & provenance

authors Lee, Gideon

provenance literature baseline

construction Two-block group-algebra code C(a,b) over C42 x C2 = SmallGroup(84, 15) (order 84), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 168; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 3, 57, 78}, b = {0, 7, 62} 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

[[168,24,8]] — two-block group-algebra code over C42 x C2

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| = 168

Parameters are also spelled out in provenance.construction of codes/168-24-8.json.

Parity checks

X-checks 84 (max weight 7) · Z-checks 84 (max weight 7)

H_X (84 checks, sparse supports)

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

H_Z (84 checks, sparse supports)

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

Code ID 168-24-8 · download JSON · raw on GitHub