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[[72,8,9]] d ≤
n
72
k
8
d
9
kd²/n
9.0
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 9, d_Z ≤ 9 · 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 9 · witness weight 9 (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, 9 qubits)
[7, 16, 20, 22, 25, 34, 42, 62, 68]
d_Z 9 · witness weight 9 (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, 9 qubits)
[5, 10, 13, 15, 22, 28, 30, 36, 59]
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 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)×36 (2,4)×126 (3,3)×36 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 36 (1,4): 36 (2,4): 126 (2,5): 432 (2,6): 180 (3,3): 36 (3,4): 144 (3,5): 900 (3,6): 3600 (3,7): 4032 (3,8): 1548 (3,9): 540 (3,10): 72
trapping sets H_Z (1,3)×36 (2,4)×126 (3,3)×36 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 36 (1,4): 36 (2,4): 126 (2,5): 432 (2,6): 180 (3,3): 36 (3,4): 144 (3,5): 900 (3,6): 3600 (3,7): 4032 (3,8): 1548 (3,9): 540 (3,10): 72

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C18 x C2 = SmallGroup(36, 5) (order 36), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 72; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 1, 3, 29}, b = {0, 8, 31} 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

[[72,8,9]] — two-block group-algebra code over C18 x C2

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| = 72

Parameters are also spelled out in provenance.construction of codes/72-8-9.json.

Parity checks

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