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[[100,2,15]] d ≤
n
100
k
2
d
15
kd²/n
4.5
w
8
X/Z
1
g
0.0028
r
6.3246
layers
2
swaps
555

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Distance

X/Z asymmetry 1 · d_X ≤ 15, d_Z ≤ 15 · 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 15 · witness weight 15 (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, 15 qubits)
[1, 11, 16, 18, 29, 34, 41, 43, 46, 56, 59, 60, 68, 86, 90]
d_Z 15 · witness weight 15 (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, 15 qubits)
[1, 2, 16, 36, 41, 45, 50, 51, 61, 69, 76, 79, 85, 86, 96]
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 8 · H_Z 8
qubit degrees H_X 4 · H_Z 4
trapping sets H_X (1,4)×100 (2,4)×100 (3,4)×200 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 100 (2,4): 100 (2,6): 1200 (3,4): 200 (3,6): 3400 (3,8): 18900 (3,10): 1750
trapping sets H_Z (1,4)×100 (2,4)×100 (3,4)×200 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 100 (2,4): 100 (2,6): 1200 (3,4): 200 (3,6): 3400 (3,8): 18900 (3,10): 1750
witness diameter X 7.0711 · Z 6.7082 (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 = 6.325
X checkZ checkqubit site (51)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 555 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 C5 x D10 = SmallGroup(50, 3) (order 50), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 100; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 1, 3, 23}, b = {0, 2, 7, 45} 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

[[100,2,15]] — two-block group-algebra code over C5 x D10

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| = 100

Parameters are also spelled out in provenance.construction of codes/100-2-15.json.

Parity checks

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