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[[168,16,16]] d ≤
n
168
k
16
d
16
kd²/n
24.381
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)
[1, 10, 12, 13, 35, 46, 54, 81, 87, 92, 97, 101, 117, 120, 122, 164]
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)
[11, 20, 33, 38, 41, 57, 61, 68, 72, 93, 99, 127, 141, 147, 152, 166]
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)×168 (2,6)×2352 (3,6)×1848 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 168 (2,6): 2352 (3,6): 1848 (3,8): 43848 (3,10): 4704
trapping sets H_Z (1,4)×168 (2,6)×2352 (3,6)×1848 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 168 (2,6): 2352 (3,6): 1848 (3,8): 43848 (3,10): 4704

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C84 = SmallGroup(84, 6) (order 84), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 168; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 10, 60, 79}, b = {0, 5, 31, 71} 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

[[168,16,16]] — two-block group-algebra code over C84

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| = 168

Parameters are also spelled out in provenance.construction of codes/168-16-16.json.

Parity checks

X-checks 84 (max weight 8) · Z-checks 84 (max weight 8)
H_X (84 checks, sparse supports)
[0, 19, 27, 51, 84, 109, 123, 162] [1, 9, 25, 32, 85, 92, 104, 165] [2, 4, 37, 61, 86, 91, 133, 166] [3, 31, 39, 63, 87, 94, 121, 135] [4, 8, 14, 38, 88, 97, 110, 152] [5, 7, 18, 42, 84, 89, 114, 167] [6, 20, 37, 44, 90, 99, 102, 116] [7, 13, 21, 45, 91, 103, 117, 157] [8, 10, 17, 41, 92, 99, 113, 154] [9, 12, 49, 72, 93, 101, 103, 145] [0, 10, 24, 48, 85, 94, 120, 160] [11, 43, 51, 74, 95, 105, 133, 147] [12, 18, 26, 50, 86, 96, 108, 122] [3, 13, 15, 22, 86, 95, 97, 159] [14, 17, 30, 54, 87, 98, 109, 126] [1, 15, 31, 55, 88, 99, 127, 163] [16, 32, 49, 56, 100, 111, 114, 128] [17, 24, 33, 57, 89, 101, 115, 129] [18, 21, 29, 53, 88, 102, 111, 125] [2, 6, 19, 28, 89, 100, 103, 142] [20, 23, 61, 80, 104, 113, 115, 156] [3, 21, 36, 60, 90, 92, 105, 132] [22, 55, 63, 81, 106, 117, 145, 158] [23, 30, 38, 62, 93, 107, 120, 134] [11, 24, 27, 34, 91, 93, 106, 108] [5, 12, 25, 35, 94, 107, 109, 148] [26, 29, 42, 66, 95, 110, 121, 138] [6, 27, 43, 67, 96, 97, 111, 139] [28, 44, 61, 68, 112, 123, 126, 140] [29, 36, 45, 69, 98, 113, 127, 141] [30, 33, 41, 65, 96, 114, 123, 137] [9, 16, 31, 40, 84, 98, 112, 115] [32, 35, 72, 83, 116, 125, 127, 164] [11, 33, 48, 71, 100, 102, 117, 144] [15, 34, 67, 74, 118, 129, 156, 165] [35, 42, 50, 73, 104, 119, 132, 146] [22, 36, 39, 46, 101, 104, 118, 120] [14, 23, 37, 47, 85, 105, 119, 121] [38, 41, 54, 76, 106, 122, 133, 150] [16, 39, 55, 77, 107, 108, 123, 151] [2, 40, 56, 72, 124, 135, 138, 152] [41, 48, 57, 78, 110, 125, 139, 153] [42, 45, 53, 75, 107, 126, 135, 149] [20, 28, 43, 52, 87, 110, 124, 127] [25, 44, 47, 80, 128, 137, 139, 167] [22, 45, 60, 79, 112, 114, 129, 155] [27, 46, 77, 81, 99, 130, 141, 164] [5, 47, 54, 62, 116, 131, 144, 157] [34, 48, 51, 58, 113, 116, 130, 132] [26, 35, 49, 59, 90, 117, 131, 133] [8, 50, 53, 66, 118, 134, 145, 160] [28, 51, 67, 82, 119, 120, 135, 161] [9, 52, 68, 80, 86, 136, 147, 150] [10, 53, 60, 69, 122, 137, 151, 162] [7, 54, 57, 65, 119, 138, 147, 159] [32, 40, 55, 64, 95, 122, 136, 139] [37, 56, 59, 83, 109, 140, 149, 151] [13, 34, 57, 71, 124, 126, 141, 163] [15, 39, 58, 82, 111, 142, 153, 167] [14, 59, 66, 73, 89, 128, 143, 155] [0, 46, 60, 63, 125, 128, 142, 144] [38, 47, 61, 70, 100, 129, 143, 145] [18, 62, 65, 76, 92, 130, 146, 156] [19, 40, 63, 77, 131, 132, 147, 166] [2, 20, 64, 83, 93, 148, 158, 160] [21, 65, 71, 78, 94, 134, 149, 161] [17, 66, 69, 75, 91, 131, 150, 158] [1, 44, 52, 67, 106, 134, 148, 151] [25, 49, 68, 70, 121, 152, 159, 161] [24, 46, 69, 79, 97, 136, 138, 153] [5, 26, 70, 76, 98, 140, 154, 163] [3, 58, 71, 74, 84, 137, 140, 155] [4, 50, 59, 72, 112, 141, 154, 156] [8, 30, 73, 75, 102, 142, 157, 164] [31, 52, 74, 82, 103, 143, 144, 158] [10, 33, 75, 79, 105, 146, 159, 166] [7, 29, 76, 78, 101, 143, 160, 165] [6, 56, 64, 77, 85, 118, 146, 161] [13, 36, 58, 78, 108, 148, 150, 162] [0, 11, 79, 81, 87, 149, 152, 163] [12, 62, 70, 80, 88, 124, 153, 164] [19, 43, 64, 81, 115, 154, 155, 165] [1, 16, 68, 82, 90, 130, 157, 166] [4, 23, 73, 83, 96, 136, 162, 167]
H_Z (84 checks, sparse supports)
[0, 5, 31, 71, 84, 94, 144, 163] [1, 10, 37, 77, 85, 99, 151, 166] [2, 12, 13, 52, 86, 103, 124, 148] [3, 14, 43, 79, 87, 97, 105, 155] [4, 15, 18, 80, 86, 88, 156, 167] [5, 17, 19, 59, 89, 109, 131, 154] [6, 21, 49, 82, 90, 103, 111, 161] [2, 7, 24, 66, 89, 91, 138, 160] [1, 8, 21, 62, 88, 92, 134, 157] [9, 23, 24, 64, 85, 93, 115, 136] [3, 10, 25, 65, 92, 94, 137, 159] [11, 13, 26, 55, 95, 108, 117, 163] [12, 27, 30, 83, 93, 96, 109, 164] [4, 13, 27, 69, 91, 97, 141, 162] [14, 29, 31, 70, 88, 98, 121, 143] [6, 8, 15, 46, 97, 99, 118, 142] [16, 19, 33, 61, 100, 115, 123, 166] [9, 17, 36, 76, 92, 98, 101, 150] [6, 18, 33, 73, 89, 96, 102, 146] [7, 9, 19, 74, 84, 103, 147, 165] [1, 20, 35, 36, 90, 104, 127, 148] [11, 21, 37, 75, 91, 102, 105, 149] [22, 24, 38, 67, 97, 106, 120, 129] [23, 25, 39, 42, 104, 107, 121, 167] [12, 24, 39, 78, 94, 101, 108, 153] [0, 14, 25, 56, 85, 109, 128, 152] [4, 26, 41, 43, 96, 110, 133, 154] [16, 18, 27, 58, 84, 108, 111, 130] [28, 31, 45, 72, 103, 112, 127, 135] [8, 20, 29, 48, 102, 110, 113, 160] [5, 16, 30, 45, 98, 107, 114, 157] [17, 20, 31, 81, 87, 99, 115, 158] [6, 32, 47, 48, 85, 100, 116, 139] [7, 22, 33, 49, 101, 114, 117, 159] [34, 36, 50, 77, 108, 118, 132, 141] [35, 37, 51, 54, 109, 116, 119, 133] [10, 23, 36, 51, 105, 113, 120, 162] [3, 26, 37, 68, 86, 90, 121, 140] [12, 38, 53, 55, 88, 107, 122, 145] [0, 28, 30, 39, 87, 120, 123, 142] [40, 43, 57, 80, 115, 124, 139, 147] [18, 32, 41, 60, 92, 114, 122, 125] [14, 28, 42, 57, 89, 110, 119, 126] [15, 29, 32, 43, 95, 111, 127, 165] [16, 44, 59, 60, 90, 112, 128, 151] [17, 34, 45, 61, 91, 113, 126, 129] [46, 48, 62, 82, 120, 130, 144, 153] [47, 49, 63, 66, 121, 128, 131, 145] [21, 35, 48, 63, 94, 117, 125, 132] [2, 11, 38, 49, 93, 100, 133, 152] [23, 50, 65, 67, 96, 119, 134, 156] [3, 40, 42, 51, 84, 95, 132, 135] [52, 55, 69, 83, 127, 136, 151, 158] [30, 44, 53, 71, 102, 126, 134, 137] [26, 40, 54, 69, 98, 122, 131, 138] [27, 41, 44, 55, 99, 106, 123, 139] [28, 56, 70, 71, 100, 124, 140, 161] [29, 46, 57, 72, 101, 125, 138, 141] [19, 58, 60, 73, 132, 142, 155, 162] [59, 61, 74, 76, 133, 140, 143, 156] [33, 47, 60, 74, 105, 129, 137, 144] [9, 22, 50, 61, 86, 104, 112, 145] [35, 62, 75, 77, 107, 131, 146, 164] [11, 52, 54, 63, 87, 106, 144, 147] [25, 64, 67, 78, 139, 148, 161, 165] [42, 56, 65, 79, 114, 138, 146, 149] [38, 52, 66, 78, 110, 134, 143, 150] [39, 53, 56, 67, 111, 118, 135, 151] [4, 40, 68, 79, 112, 136, 152, 166] [41, 58, 69, 80, 113, 137, 150, 153] [8, 70, 72, 81, 145, 152, 154, 164] [45, 59, 71, 81, 117, 141, 149, 155] [20, 34, 62, 72, 93, 116, 124, 156] [7, 47, 73, 82, 119, 143, 157, 167] [22, 64, 66, 74, 95, 118, 155, 158] [13, 54, 68, 75, 126, 150, 157, 159] [10, 50, 64, 76, 122, 146, 154, 160] [51, 65, 68, 77, 123, 130, 147, 161] [0, 53, 78, 83, 125, 149, 160, 162] [15, 57, 70, 79, 129, 153, 159, 163] [32, 46, 73, 80, 104, 128, 136, 164] [1, 34, 76, 81, 106, 130, 163, 165] [2, 63, 75, 82, 135, 142, 158, 166] [5, 44, 58, 83, 116, 140, 148, 167]
Code ID 168-16-16 · download JSON · raw on GitHub