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[[132,8,16]] d ≤
n
132
k
8
d
16
kd²/n
15.515
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 16, d_Z ≤ 16 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 16 · witness weight 16 (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, 16 qubits)
[0, 2, 8, 14, 24, 38, 48, 61, 68, 73, 87, 94, 112, 119, 128, 131]
d_Z 16 · witness weight 16 (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, 16 qubits)
[18, 24, 32, 43, 45, 50, 59, 67, 70, 71, 94, 98, 99, 102, 111, 118]
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 8 · H_Z 8
qubit degrees H_X 3–5 (mean 4.0) · H_Z 3–5 (mean 4.0)
trapping sets H_X (1,3)×66 (2,4)×198 (3,3)×88 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 66 (1,5): 66 (2,4): 198 (2,6): 1023 (2,8): 594 (3,3): 88 (3,5): 660 (3,7): 11220 (3,9): 18942 (3,11): 8382 (3,13): 462
trapping sets H_Z (1,3)×66 (2,4)×198 (3,3)×88 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 66 (1,5): 66 (2,4): 198 (2,6): 1023 (2,8): 594 (3,3): 88 (3,5): 660 (3,7): 11220 (3,9): 18942 (3,11): 8382 (3,13): 462

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C11 x S3 = SmallGroup(66, 1) (order 66), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 132; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 1, 12, 16, 59}, b = {0, 3, 14} as element indices. Check weight 8. 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

[[132,8,16]] — two-block group-algebra code over C11 x S3

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| = 132

Parameters are also spelled out in provenance.construction of codes/132-8-16.json.

Parity checks

X-checks 66 (max weight 8) · Z-checks 66 (max weight 8)
H_X (66 checks, sparse supports)
[0, 1, 29, 42, 58, 66, 74, 126] [0, 1, 19, 45, 61, 67, 77, 128] [2, 4, 35, 48, 62, 68, 69, 80] [3, 5, 15, 49, 63, 66, 69, 130] [2, 4, 25, 51, 64, 70, 71, 83] [3, 5, 26, 48, 52, 67, 71, 131] [5, 6, 9, 41, 54, 72, 73, 86] [7, 10, 21, 55, 65, 68, 73, 74] [8, 11, 22, 51, 56, 69, 74, 120] [6, 8, 9, 31, 57, 75, 76, 89] [7, 10, 32, 54, 58, 70, 76, 77] [8, 11, 12, 55, 59, 71, 77, 123] [0, 10, 12, 15, 47, 78, 79, 92] [11, 13, 16, 27, 60, 72, 79, 80] [14, 17, 28, 57, 61, 66, 73, 80] [1, 12, 14, 15, 37, 81, 82, 95] [0, 13, 16, 38, 62, 75, 82, 83] [14, 17, 18, 60, 63, 67, 76, 83] [2, 16, 18, 21, 53, 84, 85, 98] [3, 17, 19, 22, 33, 78, 85, 86] [1, 20, 23, 34, 64, 68, 79, 86] [4, 18, 20, 21, 43, 87, 88, 101] [2, 5, 19, 22, 44, 81, 88, 89] [3, 20, 23, 24, 65, 70, 82, 89] [6, 22, 24, 27, 59, 90, 91, 104] [7, 23, 25, 28, 39, 84, 91, 92] [4, 8, 26, 29, 40, 72, 85, 92] [9, 24, 26, 27, 49, 93, 94, 107] [6, 10, 25, 28, 50, 87, 94, 95] [7, 11, 26, 29, 30, 75, 88, 95] [12, 28, 30, 33, 63, 96, 97, 110] [13, 29, 31, 34, 45, 90, 97, 98] [9, 14, 32, 35, 46, 78, 91, 98] [15, 30, 32, 33, 55, 99, 100, 113] [12, 16, 31, 34, 56, 93, 100, 101] [13, 17, 32, 35, 36, 81, 94, 101] [18, 34, 36, 39, 65, 102, 103, 116] [19, 35, 37, 40, 51, 96, 103, 104] [15, 20, 38, 41, 52, 84, 97, 104] [21, 36, 38, 39, 60, 105, 106, 119] [18, 22, 37, 40, 61, 99, 106, 107] [19, 23, 38, 41, 42, 87, 100, 107] [11, 24, 40, 42, 45, 108, 109, 122] [25, 41, 43, 46, 57, 102, 109, 110] [21, 26, 44, 47, 58, 90, 103, 110] [3, 27, 42, 44, 45, 111, 112, 125] [24, 28, 43, 46, 64, 105, 112, 113] [25, 29, 44, 47, 48, 93, 106, 113] [17, 30, 46, 48, 51, 114, 115, 127] [1, 31, 47, 49, 52, 108, 115, 116] [27, 32, 50, 53, 62, 96, 109, 116] [7, 33, 48, 50, 51, 117, 118, 129] [8, 30, 34, 49, 52, 111, 118, 119] [31, 35, 50, 53, 54, 99, 112, 119] [23, 36, 52, 54, 57, 120, 121, 130] [4, 37, 53, 55, 58, 114, 121, 122] [5, 33, 38, 56, 59, 102, 115, 122] [13, 39, 54, 56, 57, 123, 124, 131] [14, 36, 40, 55, 58, 117, 124, 125] [0, 37, 41, 56, 59, 105, 118, 125] [9, 43, 59, 60, 62, 120, 126, 127] [10, 39, 44, 61, 63, 108, 121, 127] [20, 42, 46, 60, 62, 123, 128, 129] [2, 43, 47, 61, 63, 111, 124, 129] [16, 45, 50, 64, 65, 114, 126, 130] [6, 49, 53, 64, 65, 117, 128, 131]
H_Z (66 checks, sparse supports)
[0, 3, 14, 66, 67, 78, 82, 125] [1, 5, 17, 66, 67, 81, 86, 115] [2, 7, 20, 68, 70, 84, 88, 129] [2, 3, 8, 69, 71, 85, 89, 111] [4, 10, 23, 68, 70, 87, 92, 121] [4, 5, 11, 69, 71, 72, 88, 122] [6, 13, 26, 72, 75, 90, 94, 131] [6, 7, 14, 73, 76, 91, 95, 117] [0, 7, 8, 74, 75, 77, 92, 118] [9, 16, 29, 72, 75, 93, 98, 126] [9, 10, 17, 73, 76, 78, 94, 127] [1, 10, 11, 74, 77, 79, 95, 108] [12, 19, 32, 77, 78, 81, 96, 100] [12, 13, 20, 79, 82, 97, 101, 123] [2, 13, 14, 80, 81, 83, 98, 124] [15, 22, 35, 69, 78, 81, 99, 104] [15, 16, 23, 79, 82, 84, 100, 130] [4, 16, 17, 80, 83, 85, 101, 114] [18, 25, 38, 83, 84, 87, 102, 106] [18, 19, 26, 67, 85, 88, 103, 107] [6, 19, 20, 86, 87, 89, 104, 128] [21, 28, 41, 73, 84, 87, 105, 110] [21, 22, 29, 74, 85, 88, 90, 106] [9, 22, 23, 86, 89, 91, 107, 120] [24, 31, 44, 89, 90, 93, 108, 112] [24, 25, 32, 70, 91, 94, 109, 113] [12, 25, 26, 71, 92, 93, 95, 110] [27, 34, 47, 79, 90, 93, 111, 116] [27, 28, 35, 80, 91, 94, 96, 112] [15, 28, 29, 66, 92, 95, 97, 113] [30, 37, 50, 95, 96, 99, 114, 118] [30, 31, 38, 75, 97, 100, 115, 119] [18, 31, 32, 76, 98, 99, 101, 116] [33, 40, 53, 85, 96, 99, 117, 122] [33, 34, 41, 86, 97, 100, 102, 118] [21, 34, 35, 68, 98, 101, 103, 119] [36, 43, 56, 101, 102, 105, 120, 124] [36, 37, 44, 81, 103, 106, 121, 125] [24, 37, 38, 82, 104, 105, 107, 122] [39, 46, 59, 91, 102, 105, 123, 127] [39, 40, 47, 92, 103, 106, 108, 124] [27, 40, 41, 72, 104, 107, 109, 125] [42, 49, 61, 66, 107, 108, 111, 128] [42, 43, 50, 87, 109, 112, 126, 129] [30, 43, 44, 88, 110, 111, 113, 127] [45, 52, 63, 67, 97, 108, 111, 130] [45, 46, 53, 98, 109, 112, 114, 128] [33, 46, 47, 78, 110, 113, 115, 129] [48, 55, 64, 68, 71, 113, 114, 117] [48, 49, 56, 69, 93, 115, 118, 131] [36, 49, 50, 94, 116, 117, 119, 130] [51, 58, 65, 70, 74, 103, 114, 117] [51, 52, 59, 71, 104, 115, 118, 120] [39, 52, 53, 84, 116, 119, 121, 131] [8, 54, 60, 72, 76, 119, 120, 123] [54, 55, 61, 73, 77, 99, 121, 124] [42, 55, 56, 74, 100, 122, 123, 125] [11, 57, 62, 75, 80, 109, 120, 123] [57, 58, 63, 66, 76, 110, 121, 124] [45, 58, 59, 77, 90, 122, 125, 126] [0, 60, 64, 79, 83, 105, 126, 128] [48, 60, 61, 67, 80, 106, 127, 129] [1, 62, 65, 68, 82, 116, 126, 128] [51, 62, 63, 69, 83, 96, 127, 129] [3, 54, 64, 70, 86, 112, 130, 131] [5, 57, 65, 73, 89, 102, 130, 131]
Code ID 132-8-16 · download JSON · raw on GitHub