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[[84,4,12]] d ≤
n
84
k
4
d
12
kd²/n
6.857
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 12, d_Z ≤ 12 · 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 12 · witness weight 12 (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, 12 qubits)
[1, 9, 19, 23, 24, 32, 38, 40, 47, 56, 74, 83]
d_Z 12 · witness weight 12 (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, 12 qubits)
[29, 34, 49, 50, 54, 56, 59, 60, 61, 64, 67, 72]
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)×42 (2,4)×126 (3,3)×42 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 42 (1,4): 42 (2,4): 126 (2,5): 504 (2,6): 252 (3,3): 42 (3,4): 84 (3,5): 840 (3,6): 4340 (3,7): 5208 (3,8): 1848 (3,9): 756 (3,10): 168
trapping sets H_Z (1,3)×42 (2,4)×126 (3,3)×42 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 42 (1,4): 42 (2,4): 126 (2,5): 504 (2,6): 252 (3,3): 42 (3,4): 84 (3,5): 840 (3,6): 4340 (3,7): 5208 (3,8): 1848 (3,9): 756 (3,10): 168

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C42 = SmallGroup(42, 6) (order 42), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 84; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 2, 3, 22}, b = {0, 7, 21} 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 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

[[84,4,12]] — two-block group-algebra code over C42

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| = 84

Parameters are also spelled out in provenance.construction of codes/84-4-12.json.

Parity checks

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