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[[160,16,16]] d ≤
n
160
k
16
d
16
kd²/n
25.6
w
9
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 16, d_Z ≤ 16 · 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 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)
[2, 10, 12, 20, 24, 26, 38, 42, 90, 101, 106, 108, 119, 126, 143, 155]
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)
[1, 5, 53, 58, 70, 73, 85, 88, 91, 92, 101, 114, 118, 136, 139, 154]
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 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)×80 (2,6)×480 (3,6)×320 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 80 (1,5): 80 (2,6): 480 (2,7): 1600 (2,8): 800 (3,6): 320 (3,7): 720 (3,8): 5360 (3,9): 26560 (3,10): 31920 (3,11): 10640 (3,12): 3200 (3,13): 800
trapping sets H_Z (1,4)×80 (2,6)×480 (3,6)×320 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 80 (1,5): 80 (2,6): 480 (2,7): 1600 (2,8): 800 (3,6): 320 (3,7): 720 (3,8): 5360 (3,9): 26560 (3,10): 31920 (3,11): 10640 (3,12): 3200 (3,13): 800

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C40 x C2 = SmallGroup(80, 23) (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, 3, 41, 60, 65}, b = {0, 6, 56, 73} 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

[[160,16,16]] — two-block group-algebra code over C40 x C2

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| = 160

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

Parity checks

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