[[108,40,3]] · QEC Challenge

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[[108,40,3]] d =

108

kd²/n

3.333

X/Z

0.0115

4.1231

layers

swaps

157

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Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · 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 3 · witness weight 3 (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, 3 qubits)

[26, 38, 47]

d_Z 3 · witness weight 3 (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, 3 qubits)

[45, 51, 53]

certificate exact, d = 3 · CryptoMiniSat 5.14.7 SAT

X: no logical < 3 exists; Z: no logical < 3 exists

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 8 · H_Z 8

qubit degrees H_X 4 · H_Z 4

trapping sets H_X (1,4)×108 (2,4)×432 (3,0)×54 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 108 (2,4): 432 (2,6): 648 (3,0): 54 (3,4): 1674 (3,6): 5436 (3,8): 5184

trapping sets H_Z (1,4)×108 (2,4)×432 (3,0)×54 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 108 (2,4): 432 (2,6): 648 (3,0): 54 (3,4): 1674 (3,6): 5436 (3,8): 5184

witness diameter X 2.2361 · Z 1.0 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair

layout contributed by @MathysRennela · Simulated annealing over integer-grid site assignments with the repo's own research/local2d/fold_layout.search_layout, single layer at radius 4 then two layers at 7, plus a divisor scan of the two-block index convention Z_l x Z_m which gives the tighter embedding here; radius and per-site occupancy re-measured with the verifier's own rule. · 2026-10-13

X checkZ checkqubit site (56)2 qubits stacked (2 layers)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release

routing cost 157 nearest-neighbor SWAPs per round in total, at most 6 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

Construction & provenance

authors Lee, Gideon

provenance literature baseline

construction Two-block group-algebra code C(a,b) over C3 x C3 x S3 = SmallGroup(54, 12) (order 54), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 108; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 2, 23, 35}, b = {0, 2, 15, 27} 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 2bga-nonabelian (a tag, not a ranking)

locality 2D-local bilayer (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

[[108,40,3]] — two-block group-algebra code over C3 x C3 x S3

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| = 108

Parameters are also spelled out in provenance.construction of codes/108-40-3.json.

Parity checks

X-checks 54 (max weight 8) · Z-checks 54 (max weight 8)

H_X (54 checks, sparse supports)

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

H_Z (54 checks, sparse supports)

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

Code ID 108-40-3 · download JSON · raw on GitHub