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[[42,12,6]] d =
n
42
k
12
d
6
kd²/n
10.286
w
9
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 6, d_Z = 6 · w_X = 9, w_Z = 9 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 6 · witness weight 6 (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, 6 qubits)
[0, 3, 14, 20, 25, 28]
d_Z 6 · witness weight 6 (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, 6 qubits)
[6, 18, 23, 34, 35, 37]
certificate exact, d = 6 · CryptoMiniSat 5.14.7 SAT
X: no logical < 6 exists; Z: no logical < 6 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 9 · H_Z 9
qubit degrees H_X 4–5 (mean 4.5) · H_Z 4–5 (mean 4.5)
trapping sets H_X (1,4)×21 (2,4)×21 (3,3)×63 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 21 (1,5): 21 (2,4): 21 (2,5): 84 (2,6): 168 (2,7): 252 (2,8): 63 (3,3): 63 (3,4): 84 (3,5): 336 (3,6): 826 (3,7): 1449 (3,8): 2163 (3,9): 2100 (3,10): 1176 (3,11): 378 (3,12): 63
trapping sets H_Z (1,4)×21 (2,4)×21 (3,3)×63 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 21 (1,5): 21 (2,4): 21 (2,5): 84 (2,6): 168 (2,7): 252 (2,8): 63 (3,3): 63 (3,4): 84 (3,5): 336 (3,6): 826 (3,7): 1449 (3,8): 2163 (3,9): 2100 (3,10): 1176 (3,11): 378 (3,12): 63

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C21 = SmallGroup(21, 2) (order 21), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 42; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 2, 7, 12, 16}, b = {0, 1, 6, 12} as element indices. Check weight 9. 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

[[42,12,6]] — two-block group-algebra code over C21

Direction & hypothesis

Cell weight-9plus 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| = 42

Parameters are also spelled out in provenance.construction of codes/42-12-6.json.

Parity checks

X-checks 21 (max weight 9) · Z-checks 21 (max weight 9)
H_X (21 checks, sparse supports)
[0, 9, 13, 17, 18, 21, 24, 34, 40] [1, 5, 14, 15, 19, 21, 22, 36, 41] [0, 2, 12, 16, 20, 22, 23, 27, 37] [3, 7, 11, 16, 20, 22, 24, 32, 38] [1, 4, 8, 17, 18, 23, 24, 25, 39] [2, 3, 5, 15, 19, 25, 26, 30, 40] [3, 6, 10, 14, 19, 21, 25, 27, 35] [0, 4, 7, 11, 20, 26, 27, 28, 41] [1, 5, 6, 8, 18, 22, 28, 29, 33] [1, 6, 9, 13, 17, 23, 28, 30, 38] [2, 3, 7, 10, 14, 24, 29, 30, 31] [4, 8, 9, 11, 20, 25, 31, 32, 36] [0, 4, 9, 12, 16, 21, 26, 31, 33] [5, 6, 10, 13, 17, 27, 32, 33, 34] [3, 7, 11, 12, 14, 28, 34, 35, 39] [2, 7, 12, 15, 19, 23, 29, 34, 36] [0, 8, 9, 13, 16, 30, 35, 36, 37] [6, 10, 14, 15, 17, 31, 37, 38, 41] [1, 5, 10, 15, 18, 26, 32, 37, 39] [2, 11, 12, 16, 19, 33, 38, 39, 40] [4, 8, 13, 18, 20, 29, 35, 40, 41]
H_Z (21 checks, sparse supports)
[0, 1, 6, 12, 21, 23, 28, 33, 37] [1, 2, 3, 8, 22, 25, 29, 30, 39] [2, 4, 9, 15, 23, 26, 31, 36, 40] [0, 3, 4, 10, 24, 26, 27, 31, 35] [4, 5, 6, 11, 25, 28, 32, 33, 41] [5, 7, 12, 18, 22, 26, 29, 34, 39] [2, 6, 7, 13, 27, 29, 30, 34, 38] [7, 8, 9, 14, 24, 28, 31, 35, 36] [8, 10, 15, 20, 25, 29, 32, 37, 41] [5, 9, 10, 16, 21, 30, 32, 33, 37] [10, 11, 12, 17, 27, 31, 34, 38, 39] [3, 11, 13, 18, 24, 28, 32, 35, 40] [8, 12, 13, 19, 23, 33, 35, 36, 40] [0, 13, 14, 15, 21, 30, 34, 37, 41] [6, 14, 16, 20, 22, 27, 31, 35, 38] [1, 11, 15, 16, 22, 26, 36, 38, 39] [2, 16, 17, 18, 23, 24, 33, 37, 40] [3, 9, 17, 19, 21, 25, 30, 34, 38] [4, 14, 18, 19, 21, 25, 29, 39, 41] [0, 5, 19, 20, 22, 26, 27, 36, 40] [1, 7, 17, 20, 23, 24, 28, 32, 41]
Code ID 42-12-6 · download JSON · raw on GitHub