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[[102,16,11]] d ≤
n
102
k
16
d
11
kd²/n
18.98
w
9
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 11, d_Z ≤ 11 · 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 11 · witness weight 11 (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, 11 qubits)
[7, 15, 16, 28, 44, 62, 70, 82, 90, 92, 101]
d_Z 11 · witness weight 11 (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, 11 qubits)
[11, 20, 26, 29, 31, 38, 49, 50, 69, 81, 83]
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 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)×51 (2,4)×51 (3,4)×102 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 51 (1,5): 51 (2,4): 51 (2,5): 51 (2,6): 306 (2,7): 918 (2,8): 306 (3,4): 102 (3,5): 153 (3,6): 510 (3,7): 2550 (3,8): 5151 (3,9): 12291 (3,10): 12801 (3,11): 2907 (3,12): 1020 (3,13): 102
trapping sets H_Z (1,4)×51 (2,4)×51 (3,4)×102 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 51 (1,5): 51 (2,4): 51 (2,5): 51 (2,6): 306 (2,7): 918 (2,8): 306 (3,4): 102 (3,5): 153 (3,6): 510 (3,7): 2550 (3,8): 5151 (3,9): 12291 (3,10): 12801 (3,11): 2907 (3,12): 1020 (3,13): 102

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C51 = SmallGroup(51, 1) (order 51), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 102; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 2, 11, 19, 31}, b = {0, 7, 21, 37} 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

[[102,16,11]] — two-block group-algebra code over C51

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| = 102

Parameters are also spelled out in provenance.construction of codes/102-16-11.json.

Parity checks

X-checks 51 (max weight 9) · Z-checks 51 (max weight 9)
H_X (51 checks, sparse supports)
[0, 24, 36, 38, 47, 51, 69, 85, 99] [1, 20, 32, 40, 49, 52, 65, 87, 95] [0, 2, 27, 39, 41, 53, 72, 88, 101] [3, 22, 34, 42, 50, 54, 67, 83, 97] [1, 4, 23, 35, 43, 55, 68, 90, 98] [2, 5, 30, 42, 44, 54, 56, 75, 91] [3, 6, 25, 37, 45, 57, 70, 86, 100] [4, 7, 26, 38, 46, 51, 58, 71, 93] [5, 8, 33, 45, 47, 57, 59, 78, 94] [6, 9, 28, 40, 48, 52, 60, 73, 89] [7, 10, 29, 41, 49, 53, 61, 74, 96] [0, 8, 11, 36, 48, 60, 62, 81, 97] [9, 12, 31, 43, 50, 55, 63, 76, 92] [1, 10, 13, 32, 44, 56, 64, 77, 99] [2, 11, 14, 39, 50, 63, 65, 84, 100] [3, 12, 15, 34, 46, 58, 66, 79, 95] [4, 13, 16, 35, 47, 59, 67, 80, 101] [3, 5, 14, 17, 42, 52, 66, 68, 87] [6, 15, 18, 37, 49, 61, 69, 82, 98] [0, 7, 16, 19, 38, 54, 62, 70, 83] [6, 8, 17, 20, 45, 55, 69, 71, 90] [1, 9, 18, 21, 40, 51, 64, 72, 85] [2, 10, 19, 22, 41, 57, 65, 73, 86] [9, 11, 20, 23, 48, 58, 72, 74, 93] [4, 12, 21, 24, 43, 53, 67, 75, 88] [5, 13, 22, 25, 44, 60, 68, 76, 89] [12, 14, 23, 26, 50, 61, 75, 77, 96] [7, 15, 24, 27, 46, 56, 70, 78, 91] [8, 16, 25, 28, 47, 63, 71, 79, 92] [3, 15, 17, 26, 29, 64, 78, 80, 99] [10, 18, 27, 30, 49, 59, 73, 81, 94] [0, 11, 19, 28, 31, 66, 74, 82, 95] [6, 18, 20, 29, 32, 67, 81, 83, 101] [1, 13, 21, 30, 33, 62, 76, 84, 97] [2, 14, 22, 31, 34, 69, 77, 85, 98] [9, 21, 23, 32, 35, 54, 70, 84, 86] [4, 16, 24, 33, 36, 65, 79, 87, 100] [5, 17, 25, 34, 37, 51, 72, 80, 88] [12, 24, 26, 35, 38, 57, 73, 87, 89] [7, 19, 27, 36, 39, 52, 68, 82, 90] [8, 20, 28, 37, 40, 53, 75, 83, 91] [15, 27, 29, 38, 41, 60, 76, 90, 92] [10, 22, 30, 39, 42, 55, 71, 85, 93] [11, 23, 31, 40, 43, 56, 78, 86, 94] [18, 30, 32, 41, 44, 63, 79, 93, 95] [13, 25, 33, 42, 45, 58, 74, 88, 96] [14, 26, 34, 43, 46, 59, 81, 89, 97] [21, 33, 35, 44, 47, 66, 82, 96, 98] [16, 28, 36, 45, 48, 61, 77, 91, 99] [17, 29, 37, 46, 49, 62, 84, 92, 100] [19, 31, 39, 48, 50, 64, 80, 94, 101]
H_Z (51 checks, sparse supports)
[0, 7, 21, 37, 51, 53, 62, 70, 82] [1, 9, 17, 39, 52, 55, 64, 72, 84] [2, 10, 24, 40, 53, 56, 65, 73, 85] [3, 5, 19, 35, 54, 57, 66, 68, 80] [4, 12, 20, 42, 55, 58, 67, 75, 87] [5, 13, 27, 43, 56, 59, 68, 76, 88] [6, 8, 22, 38, 57, 60, 69, 71, 83] [7, 15, 23, 45, 58, 61, 70, 78, 90] [8, 16, 30, 46, 59, 62, 71, 79, 91] [9, 11, 25, 41, 60, 63, 72, 74, 86] [10, 18, 26, 48, 61, 64, 73, 81, 93] [11, 19, 33, 49, 62, 65, 74, 82, 94] [12, 14, 28, 44, 63, 66, 75, 77, 89] [13, 21, 29, 50, 64, 67, 76, 84, 96] [1, 14, 22, 36, 65, 68, 77, 85, 97] [15, 17, 31, 47, 66, 69, 78, 80, 92] [3, 16, 24, 32, 67, 70, 79, 87, 99] [4, 17, 25, 39, 68, 71, 80, 88, 100] [0, 18, 20, 34, 69, 72, 81, 83, 95] [6, 19, 27, 35, 70, 73, 82, 90, 101] [7, 20, 28, 42, 52, 71, 74, 83, 91] [2, 21, 23, 37, 72, 75, 84, 86, 98] [9, 22, 30, 38, 54, 73, 76, 85, 93] [10, 23, 31, 45, 55, 74, 77, 86, 94] [5, 24, 26, 40, 51, 75, 78, 87, 89] [12, 25, 33, 41, 57, 76, 79, 88, 96] [13, 26, 34, 48, 58, 77, 80, 89, 97] [8, 27, 29, 43, 53, 78, 81, 90, 92] [15, 28, 36, 44, 60, 79, 82, 91, 99] [16, 29, 37, 50, 61, 80, 83, 92, 100] [11, 30, 32, 46, 56, 81, 84, 93, 95] [18, 31, 39, 47, 63, 82, 85, 94, 101] [3, 19, 32, 40, 52, 64, 83, 86, 95] [14, 33, 35, 49, 59, 84, 87, 96, 98] [0, 21, 34, 42, 54, 66, 85, 88, 97] [6, 22, 35, 43, 55, 67, 86, 89, 98] [1, 17, 36, 38, 51, 62, 87, 90, 99] [2, 24, 37, 45, 57, 69, 88, 91, 100] [9, 25, 38, 46, 51, 58, 70, 89, 92] [4, 20, 39, 41, 53, 65, 90, 93, 101] [5, 27, 40, 48, 52, 60, 72, 91, 94] [12, 28, 41, 49, 53, 61, 73, 92, 95] [7, 23, 42, 44, 54, 56, 68, 93, 96] [8, 30, 43, 50, 55, 63, 75, 94, 97] [1, 15, 31, 44, 56, 64, 76, 95, 98] [10, 26, 45, 47, 57, 59, 71, 96, 99] [3, 11, 33, 46, 58, 66, 78, 97, 100] [4, 18, 34, 47, 51, 59, 67, 79, 98] [0, 13, 29, 48, 60, 62, 74, 99, 101] [6, 14, 36, 49, 52, 61, 69, 81, 100] [2, 16, 32, 50, 54, 63, 65, 77, 101]
Code ID 102-16-11 · download JSON · raw on GitHub