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[[84,22,6]] d =
n
84
k
22
d
6
kd²/n
9.429
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)
[7, 18, 24, 33, 42, 72]
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)
[7, 13, 42, 61, 65, 67]
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)×42 (2,5)×126 (3,5)×252 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 42 (1,5): 42 (2,5): 126 (2,6): 252 (2,7): 588 (2,8): 420 (3,5): 252 (3,6): 546 (3,7): 2184 (3,8): 4788 (3,9): 9422 (3,10): 9576 (3,11): 546 (3,12): 798 (3,13): 420
trapping sets H_Z (1,4)×42 (2,5)×126 (3,5)×252 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 42 (1,5): 42 (2,5): 126 (2,6): 252 (2,7): 588 (2,8): 420 (3,5): 252 (3,6): 546 (3,7): 2184 (3,8): 4788 (3,9): 9422 (3,10): 9576 (3,11): 546 (3,12): 798 (3,13): 420

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C7 x S3 = SmallGroup(42, 3) (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, 12, 36, 40}, b = {0, 2, 25, 33} 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 2bga-nonabelian (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

[[84,22,6]] — two-block group-algebra code over C7 x S3

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| = 84

Parameters are also spelled out in provenance.construction of codes/84-22-6.json.

Parity checks

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