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[[162,12,17]] d ≤
n
162
k
12
d
17
kd²/n
21.407
w
9
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 17, d_Z ≤ 17 · 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 17 · witness weight 17 (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, 17 qubits)
[6, 13, 22, 23, 42, 59, 69, 73, 79, 101, 108, 110, 112, 117, 140, 150, 152]
d_Z 17 · witness weight 17 (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, 17 qubits)
[18, 19, 26, 34, 49, 51, 54, 76, 90, 95, 105, 111, 118, 128, 130, 135, 156]
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)×81 (2,5)×81 (3,5)×81 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 81 (1,5): 81 (2,5): 81 (2,6): 486 (2,7): 1458 (2,8): 810 (3,5): 81 (3,6): 432 (3,7): 2673 (3,8): 6561 (3,9): 21222 (3,10): 29160 (3,11): 10773 (3,12): 2592 (3,13): 810
trapping sets H_Z (1,4)×81 (2,5)×81 (3,5)×81 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 81 (1,5): 81 (2,5): 81 (2,6): 486 (2,7): 1458 (2,8): 810 (3,5): 81 (3,6): 432 (3,7): 2673 (3,8): 6561 (3,9): 21222 (3,10): 29160 (3,11): 10773 (3,12): 2592 (3,13): 810

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C9 x C3 x C3 = SmallGroup(81, 11) (order 81), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 162; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 8, 24, 28, 80}, b = {0, 3, 21, 77} 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

[[162,12,17]] — two-block group-algebra code over C9 x C3 x C3

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| = 162

Parameters are also spelled out in provenance.construction of codes/162-12-17.json.

Parity checks

X-checks 81 (max weight 9) · Z-checks 81 (max weight 9)
H_X (81 checks, sparse supports)
[0, 17, 19, 25, 26, 81, 93, 120, 137] [1, 14, 32, 38, 39, 82, 84, 102, 133] [2, 4, 33, 37, 44, 83, 107, 138, 151] [2, 3, 35, 39, 45, 81, 84, 87, 152] [4, 36, 40, 46, 47, 85, 97, 110, 141] [5, 23, 27, 51, 52, 86, 88, 115, 128] [6, 8, 28, 50, 57, 87, 91, 120, 147] [6, 7, 30, 52, 58, 82, 88, 93, 96] [0, 8, 53, 59, 60, 89, 94, 123, 150] [7, 9, 11, 12, 51, 90, 102, 125, 158] [9, 10, 13, 53, 57, 83, 91, 99, 159] [11, 14, 54, 58, 63, 92, 113, 128, 153] [6, 10, 12, 55, 63, 84, 93, 100, 117] [11, 13, 56, 60, 64, 85, 94, 101, 115] [5, 9, 14, 61, 65, 95, 116, 130, 155] [15, 17, 41, 45, 66, 96, 100, 133, 144] [15, 16, 43, 47, 67, 86, 92, 97, 102] [1, 17, 48, 68, 69, 98, 103, 136, 146] [16, 18, 20, 21, 46, 99, 105, 115, 138] [18, 19, 22, 48, 66, 87, 100, 107, 112] [2, 20, 23, 67, 72, 101, 108, 141, 157] [15, 19, 21, 49, 72, 81, 88, 102, 113] [3, 20, 22, 69, 73, 89, 103, 110, 114] [4, 18, 23, 70, 74, 104, 111, 143, 159] [0, 21, 24, 27, 67, 82, 90, 105, 161] [22, 25, 28, 29, 68, 106, 123, 131, 144] [18, 24, 26, 29, 69, 91, 107, 118, 135] [25, 27, 30, 70, 72, 92, 108, 119, 133] [0, 15, 28, 73, 75, 109, 134, 146, 160] [20, 27, 29, 71, 75, 86, 94, 110, 121] [16, 24, 28, 30, 74, 95, 111, 122, 136] [13, 31, 33, 34, 59, 110, 112, 118, 147] [31, 32, 35, 61, 63, 96, 106, 113, 120] [6, 33, 36, 64, 76, 114, 121, 150, 156] [11, 32, 34, 62, 76, 82, 97, 108, 115] [7, 33, 35, 65, 77, 98, 109, 116, 123] [8, 10, 31, 36, 78, 107, 117, 124, 152] [1, 34, 37, 40, 64, 86, 99, 118, 125] [9, 35, 38, 41, 42, 119, 126, 136, 153] [31, 37, 39, 42, 65, 83, 100, 120, 131] [10, 38, 40, 43, 76, 101, 121, 128, 132] [1, 11, 41, 77, 79, 122, 129, 155, 161] [12, 33, 40, 42, 79, 85, 103, 123, 134] [13, 37, 41, 43, 78, 104, 124, 130, 135] [1, 3, 44, 47, 76, 88, 105, 125, 149] [4, 42, 45, 48, 77, 89, 106, 126, 147] [2, 31, 43, 46, 49, 127, 143, 148, 156] [38, 45, 47, 49, 78, 96, 108, 128, 139] [3, 32, 46, 48, 79, 109, 129, 140, 150] [34, 41, 44, 48, 49, 98, 111, 130, 142] [5, 29, 50, 53, 73, 85, 112, 131, 138] [18, 30, 51, 54, 55, 130, 132, 139, 157] [25, 50, 52, 55, 74, 87, 113, 126, 133] [19, 51, 53, 56, 75, 114, 127, 134, 141] [5, 20, 24, 54, 80, 125, 135, 142, 159] [21, 28, 53, 55, 80, 89, 116, 129, 136] [22, 26, 50, 54, 56, 83, 117, 137, 143] [5, 7, 57, 60, 75, 90, 97, 118, 138] [8, 24, 55, 58, 61, 98, 119, 139, 144] [6, 25, 56, 59, 62, 140, 145, 152, 160] [26, 51, 58, 60, 62, 92, 121, 141, 148] [7, 27, 59, 61, 80, 122, 142, 146, 149] [29, 54, 57, 61, 62, 95, 124, 143, 151] [8, 13, 63, 65, 80, 103, 112, 126, 144] [10, 14, 50, 62, 64, 104, 127, 145, 157] [12, 52, 59, 64, 65, 114, 129, 146, 154] [4, 16, 66, 69, 79, 94, 99, 131, 147] [17, 37, 49, 67, 70, 95, 132, 148, 153] [3, 15, 38, 68, 71, 93, 149, 154, 161] [39, 46, 67, 69, 71, 101, 134, 145, 150] [16, 40, 44, 68, 70, 90, 135, 151, 155] [2, 42, 66, 70, 71, 91, 104, 137, 152] [17, 22, 44, 72, 74, 106, 116, 139, 153] [19, 23, 45, 71, 73, 117, 140, 154, 156] [21, 47, 68, 73, 74, 109, 142, 155, 158] [23, 26, 30, 66, 75, 124, 132, 145, 156] [14, 35, 57, 76, 78, 111, 119, 148, 157] [12, 32, 36, 58, 77, 81, 149, 158, 160] [9, 34, 60, 77, 78, 105, 122, 151, 159] [36, 39, 43, 63, 79, 127, 137, 154, 160] [0, 52, 56, 72, 80, 84, 140, 158, 161]
H_Z (81 checks, sparse supports)
[0, 3, 21, 77, 81, 89, 105, 109, 161] [1, 7, 24, 34, 82, 98, 118, 122, 125] [2, 10, 39, 56, 83, 84, 101, 127, 152] [1, 3, 12, 80, 84, 103, 125, 129, 149] [4, 13, 42, 50, 83, 85, 104, 126, 147] [5, 16, 29, 37, 86, 95, 131, 135, 138] [3, 6, 19, 52, 87, 88, 93, 114, 140] [5, 7, 21, 44, 88, 90, 116, 138, 142] [8, 22, 45, 55, 87, 89, 117, 139, 144] [9, 24, 57, 70, 90, 91, 95, 119, 159] [6, 10, 26, 71, 91, 93, 117, 121, 145] [11, 16, 27, 60, 90, 92, 94, 115, 122] [0, 7, 12, 68, 90, 93, 123, 146, 158] [8, 13, 29, 66, 91, 94, 112, 124, 144] [14, 30, 62, 67, 82, 92, 95, 145, 157] [7, 15, 32, 47, 96, 97, 102, 109, 149] [4, 16, 34, 57, 97, 99, 111, 147, 151] [17, 35, 49, 58, 81, 96, 98, 148, 153] [10, 18, 37, 66, 99, 100, 104, 107, 132] [12, 15, 19, 39, 81, 100, 102, 134, 154] [13, 20, 40, 69, 99, 101, 103, 110, 135] [1, 9, 16, 21, 99, 102, 105, 136, 155] [17, 22, 42, 63, 100, 103, 106, 137, 153] [23, 43, 64, 71, 86, 101, 104, 154, 156] [18, 24, 44, 78, 105, 107, 111, 135, 139] [25, 32, 45, 72, 81, 106, 108, 133, 140] [2, 19, 26, 36, 81, 107, 137, 141, 156] [20, 27, 34, 47, 86, 105, 108, 110, 142] [28, 35, 48, 74, 87, 106, 109, 111, 136] [4, 22, 29, 31, 106, 107, 110, 131, 143] [23, 30, 49, 76, 88, 108, 111, 132, 156] [19, 31, 50, 63, 112, 113, 117, 120, 127] [11, 21, 32, 52, 82, 113, 115, 129, 158] [22, 33, 53, 65, 83, 112, 114, 116, 123] [5, 13, 18, 34, 112, 115, 118, 130, 159] [14, 35, 55, 72, 84, 113, 116, 119, 157] [12, 36, 56, 73, 85, 114, 117, 158, 160] [26, 31, 37, 57, 83, 118, 120, 124, 148] [27, 38, 58, 76, 82, 119, 121, 128, 149] [0, 6, 32, 39, 82, 84, 120, 150, 160] [29, 33, 40, 60, 85, 118, 121, 123, 151] [30, 41, 61, 78, 96, 119, 122, 124, 130] [8, 25, 35, 42, 119, 120, 123, 126, 152] [36, 43, 62, 75, 97, 121, 124, 127, 160] [9, 37, 44, 54, 83, 125, 130, 151, 153] [38, 45, 52, 63, 84, 96, 126, 128, 154] [46, 53, 64, 79, 85, 99, 127, 129, 150] [5, 11, 40, 47, 85, 97, 125, 128, 155] [41, 48, 55, 65, 98, 100, 126, 129, 130] [14, 43, 49, 51, 102, 127, 128, 130, 148] [25, 39, 50, 66, 87, 131, 133, 137, 145] [40, 51, 67, 75, 86, 90, 132, 134, 141] [1, 15, 27, 52, 86, 88, 133, 146, 161] [28, 42, 53, 69, 89, 91, 131, 134, 136] [26, 43, 54, 70, 92, 132, 135, 137, 143] [17, 30, 38, 55, 93, 132, 133, 136, 139] [0, 56, 71, 79, 94, 134, 137, 140, 161] [2, 18, 50, 57, 87, 91, 138, 143, 157] [47, 51, 58, 72, 88, 92, 139, 141, 158] [48, 59, 73, 80, 89, 112, 140, 142, 146] [4, 20, 53, 60, 89, 94, 138, 141, 159] [49, 54, 61, 74, 95, 113, 139, 142, 143] [23, 46, 56, 62, 115, 140, 141, 143, 145] [15, 25, 58, 63, 92, 93, 113, 144, 160] [59, 64, 69, 75, 94, 114, 118, 145, 146] [17, 28, 61, 65, 95, 116, 120, 144, 146] [6, 31, 45, 66, 96, 100, 147, 152, 156] [46, 60, 67, 76, 97, 101, 105, 148, 150] [44, 61, 68, 77, 98, 106, 149, 151, 155] [8, 33, 48, 69, 98, 103, 107, 147, 150] [2, 62, 70, 78, 104, 108, 148, 151, 152] [3, 36, 59, 71, 110, 149, 150, 152, 154] [11, 38, 67, 72, 101, 102, 108, 153, 161] [65, 68, 73, 79, 103, 109, 131, 154, 155] [14, 41, 70, 74, 104, 111, 133, 153, 155] [33, 46, 73, 75, 109, 110, 134, 138, 156] [20, 51, 64, 76, 114, 115, 121, 125, 157] [9, 74, 77, 80, 116, 122, 126, 158, 159] [10, 23, 54, 78, 117, 124, 128, 157, 159] [28, 59, 77, 79, 122, 123, 129, 147, 160] [24, 41, 68, 80, 135, 136, 142, 144, 161]
Code ID 162-12-17 · download JSON · raw on GitHub