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[[126,6,14]] d ≤
n
126
k
6
d
14
kd²/n
9.333
w
6
X/Z
1
g
0.0081
r
5.831
layers
2
swaps
609

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Distance

X/Z asymmetry 1 · d_X ≤ 14, d_Z ≤ 14 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 14 · witness weight 14 (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, 14 qubits)
[4, 18, 21, 33, 50, 52, 58, 71, 79, 88, 105, 116, 119, 121]
d_Z 14 · witness weight 14 (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, 14 qubits)
[1, 5, 9, 23, 27, 32, 48, 81, 92, 94, 96, 102, 118, 123]
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 6 · H_Z 6
qubit degrees H_X 3 · H_Z 3
trapping sets H_X (1,3)×126 (2,4)×945 (3,3)×189 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 126 (2,4): 945 (3,3): 189 (3,5): 8883 (3,7): 1260
trapping sets H_Z (1,3)×126 (2,4)×945 (3,3)×189 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 126 (2,4): 945 (3,3): 189 (3,5): 8883 (3,7): 1260
witness diameter X 8.6023 · Z 9.2195 (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
r = 5.831
X checkZ checkqubit site (64)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 609 nearest-neighbor SWAPs per round in total, at most 9 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 C63 = SmallGroup(63, 2) (order 63), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 126; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 2, 21}, b = {0, 10, 41} as element indices. Check weight 6. 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 2D-local bilayer (computed from the layout)
weight class weight ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[126,6,14]] — two-block group-algebra code over C63

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| = 126

Parameters are also spelled out in provenance.construction of codes/126-6-14.json.

Parity checks

X-checks 63 (max weight 6) · Z-checks 63 (max weight 6)
H_X (63 checks, sparse supports)
[0, 42, 55, 63, 99, 124] [1, 24, 48, 64, 97, 125] [0, 2, 60, 65, 77, 108] [3, 36, 51, 66, 109, 111] [4, 30, 53, 67, 103, 105] [1, 5, 33, 68, 83, 106] [6, 34, 56, 69, 107, 116] [2, 7, 62, 70, 86, 116] [3, 8, 45, 64, 71, 117] [9, 46, 58, 72, 118, 119] [4, 10, 39, 63, 73, 112] [11, 40, 59, 74, 113, 114] [5, 12, 42, 75, 92, 114] [6, 13, 43, 67, 76, 115] [14, 44, 61, 77, 87, 122] [7, 15, 20, 67, 78, 95] [8, 16, 53, 68, 79, 122] [9, 17, 54, 69, 80, 123] [10, 18, 48, 65, 81, 119] [11, 19, 49, 66, 82, 120] [20, 50, 62, 83, 93, 121] [0, 12, 21, 66, 84, 101] [13, 22, 51, 73, 85, 121] [14, 23, 52, 74, 86, 96] [15, 24, 29, 73, 87, 104] [4, 16, 25, 74, 75, 88] [17, 26, 59, 76, 89, 125] [1, 18, 27, 69, 70, 90] [19, 28, 56, 71, 91, 124] [20, 29, 57, 72, 92, 102] [2, 21, 30, 71, 93, 110] [3, 22, 31, 72, 81, 94] [23, 32, 58, 82, 95, 105] [24, 33, 38, 81, 96, 113] [10, 25, 34, 82, 84, 97] [11, 26, 35, 83, 85, 98] [5, 27, 36, 76, 78, 99] [6, 28, 37, 77, 79, 100] [29, 38, 61, 80, 101, 111] [7, 30, 39, 79, 102, 118] [8, 31, 40, 80, 90, 103] [9, 32, 41, 63, 91, 104] [33, 42, 47, 90, 105, 120] [18, 34, 43, 91, 93, 106] [19, 35, 44, 92, 94, 107] [12, 36, 45, 85, 87, 108] [13, 37, 46, 86, 88, 109] [14, 38, 47, 64, 89, 110] [15, 39, 48, 88, 111, 123] [16, 40, 49, 89, 99, 112] [17, 41, 50, 65, 100, 113] [27, 43, 51, 100, 102, 114] [28, 44, 52, 101, 103, 115] [21, 45, 53, 94, 96, 116] [22, 46, 54, 95, 97, 117] [23, 47, 55, 68, 98, 118] [25, 49, 56, 98, 108, 119] [26, 50, 57, 70, 109, 120] [37, 52, 58, 110, 112, 121] [31, 54, 59, 104, 106, 122] [32, 55, 60, 75, 107, 123] [35, 57, 61, 78, 117, 124] [41, 60, 62, 84, 115, 125]
H_Z (63 checks, sparse supports)
[0, 10, 41, 63, 65, 84] [1, 8, 47, 64, 68, 90] [2, 18, 50, 65, 70, 93] [3, 19, 21, 66, 71, 94] [4, 13, 15, 67, 73, 88] [5, 16, 55, 68, 75, 99] [6, 17, 27, 69, 76, 100] [7, 27, 57, 70, 78, 102] [8, 28, 30, 71, 79, 103] [9, 29, 31, 72, 80, 104] [10, 22, 24, 73, 81, 97] [11, 23, 25, 74, 82, 98] [12, 25, 60, 75, 84, 108] [13, 26, 36, 76, 85, 109] [2, 14, 37, 77, 86, 110] [15, 36, 61, 78, 87, 111] [16, 37, 39, 79, 88, 112] [17, 38, 40, 80, 89, 113] [18, 31, 33, 81, 90, 106] [19, 32, 34, 82, 91, 107] [5, 20, 35, 78, 83, 92] [21, 34, 62, 84, 93, 116] [22, 35, 45, 85, 94, 117] [7, 23, 46, 86, 95, 118] [14, 24, 45, 64, 87, 96] [25, 46, 48, 88, 97, 119] [26, 47, 49, 89, 98, 120] [27, 40, 42, 90, 99, 114] [28, 41, 43, 91, 100, 115] [12, 29, 44, 87, 92, 101] [20, 30, 43, 67, 93, 102] [31, 44, 53, 94, 103, 122] [15, 32, 54, 95, 104, 123] [23, 33, 53, 68, 96, 105] [1, 34, 54, 69, 97, 106] [35, 55, 56, 98, 107, 124] [0, 36, 49, 66, 99, 108] [37, 50, 51, 100, 109, 121] [21, 38, 52, 96, 101, 110] [29, 39, 51, 73, 102, 111] [4, 40, 52, 74, 103, 112] [24, 41, 59, 104, 113, 125] [4, 32, 42, 63, 75, 105] [5, 43, 59, 76, 106, 114] [6, 44, 60, 77, 107, 115] [2, 45, 56, 71, 108, 116] [3, 46, 57, 72, 109, 117] [30, 47, 58, 105, 110, 118] [3, 38, 48, 64, 81, 111] [10, 49, 58, 82, 112, 119] [11, 33, 50, 83, 113, 120] [11, 12, 51, 66, 85, 114] [13, 52, 62, 86, 115, 121] [6, 7, 53, 67, 79, 116] [8, 54, 61, 80, 117, 122] [9, 39, 55, 63, 118, 123] [9, 18, 56, 69, 91, 119] [19, 42, 57, 92, 120, 124] [20, 22, 58, 72, 95, 121] [14, 16, 59, 74, 89, 122] [17, 48, 60, 65, 123, 125] [0, 28, 61, 77, 101, 124] [1, 26, 62, 70, 83, 125]
Code ID 126-6-14 · download JSON · raw on GitHub