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[[112,6,14]] d ≤
n
112
k
6
d
14
kd²/n
10.5
w
7
X/Z
1
g
0.0062
r
6.4031
layers
2
swaps
707

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Distance

X/Z asymmetry 1 · d_X ≤ 14, d_Z ≤ 14 · 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 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)
[9, 17, 26, 27, 29, 50, 52, 54, 57, 63, 72, 78, 94, 108]
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)
[6, 13, 14, 29, 50, 54, 64, 65, 69, 72, 90, 98, 101, 106]
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)×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 (1,4): 56 (2,4): 168 (2,5): 672 (2,6): 336 (3,3): 56 (3,4): 168 (3,5): 784 (3,6): 5488 (3,7): 7952 (3,8): 2856 (3,9): 1008 (3,10): 224
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
(1,3): 56 (1,4): 56 (2,4): 168 (2,5): 672 (2,6): 336 (3,3): 56 (3,4): 168 (3,5): 784 (3,6): 5488 (3,7): 7952 (3,8): 2856 (3,9): 1008 (3,10): 224
witness diameter X 7.6158 · Z 7.6158 (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 = 6.403
X checkZ checkqubit site (58)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 707 nearest-neighbor SWAPs per round in total, at most 11 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 C56 = SmallGroup(56, 2) (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, 14, 53}, b = {0, 2, 25} 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 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

[[112,6,14]] — two-block group-algebra code over C56

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| = 112

Parameters are also spelled out in provenance.construction of codes/112-6-14.json.

Parity checks

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