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[[160,14,16]] d ≤
n
160
k
14
d
16
kd²/n
22.4
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)
[21, 36, 50, 51, 55, 56, 72, 77, 79, 83, 88, 94, 95, 101, 136, 151]
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)
[20, 28, 30, 42, 57, 65, 74, 90, 108, 109, 110, 115, 121, 132, 135, 157]
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 6 · H_Z 6 (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 4 · H_Z 4
trapping sets H_X (1,4)×160 (2,6)×2240 (3,6)×1120 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 160 (2,6): 2240 (3,6): 1120 (3,8): 43680 (3,10): 4480
trapping sets H_Z (1,4)×160 (2,6)×2240 (3,6)×1120 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 160 (2,6): 2240 (3,6): 1120 (3,8): 43680 (3,10): 4480

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C80 = SmallGroup(80, 2) (order 80), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 160; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 1, 49, 73}, b = {0, 7, 28, 38} 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 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

[[160,14,16]] — two-block group-algebra code over C80

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| = 160

Parameters are also spelled out in provenance.construction of codes/160-14-16.json.

Parity checks

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