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[[90,24,6]] d =
n
90
k
24
d
6
kd²/n
9.6
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)
[3, 9, 24, 40, 60, 79]
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, 16, 33, 40, 61, 71]
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 3–6 (mean 4.5) · H_Z 3–6 (mean 4.5)
trapping sets H_X (1,3)×45 (2,4)×135 (3,3)×45 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 45 (1,6): 45 (2,4): 135 (2,6): 45 (2,7): 810 (2,10): 540 (3,3): 45 (3,5): 585 (3,6): 585 (3,7): 495 (3,8): 8055 (3,9): 2340 (3,10): 1170 (3,11): 15075 (3,12): 1950 (3,13): 1620 (3,14): 5580 (3,16): 450
trapping sets H_Z (1,3)×45 (2,4)×135 (3,3)×45 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 45 (1,6): 45 (2,4): 135 (2,6): 45 (2,7): 810 (2,10): 540 (3,3): 45 (3,5): 585 (3,6): 585 (3,7): 495 (3,8): 8055 (3,9): 2340 (3,10): 1170 (3,11): 15075 (3,12): 1950 (3,13): 1620 (3,14): 5580 (3,16): 450

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C45 = SmallGroup(45, 1) (order 45), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 90; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 1, 2, 6, 14, 25}, b = {0, 7, 17} 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

[[90,24,6]] — two-block group-algebra code over C45

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| = 90

Parameters are also spelled out in provenance.construction of codes/90-24-6.json.

Parity checks

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