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[[170,16,17]] d ≤
n
170
k
16
d
17
kd²/n
27.2
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)
[0, 2, 5, 25, 57, 60, 67, 80, 83, 100, 102, 110, 112, 126, 136, 148, 168]
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)
[10, 19, 51, 55, 56, 70, 76, 78, 82, 83, 104, 124, 136, 144, 155, 157, 169]
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)×85 (2,6)×510 (3,6)×595 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 85 (1,5): 85 (2,6): 510 (2,7): 1700 (2,8): 850 (3,6): 595 (3,7): 595 (3,8): 4930 (3,9): 28645 (3,10): 33915 (3,11): 11560 (3,12): 3400 (3,13): 850
trapping sets H_Z (1,4)×85 (2,6)×510 (3,6)×595 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 85 (1,5): 85 (2,6): 510 (2,7): 1700 (2,8): 850 (3,6): 595 (3,7): 595 (3,8): 4930 (3,9): 28645 (3,10): 33915 (3,11): 11560 (3,12): 3400 (3,13): 850

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C85 = SmallGroup(85, 1) (order 85), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 170; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 2, 24, 34, 53}, b = {0, 1, 18, 80} 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

[[170,16,17]] — two-block group-algebra code over C85

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| = 170

Parameters are also spelled out in provenance.construction of codes/170-16-17.json.

Parity checks

X-checks 85 (max weight 9) · Z-checks 85 (max weight 9)
H_X (85 checks, sparse supports)
[0, 39, 40, 49, 74, 85, 95, 97, 160] [1, 24, 43, 53, 78, 85, 86, 101, 144] [0, 2, 44, 45, 54, 87, 100, 102, 164] [3, 28, 47, 57, 81, 86, 88, 105, 148] [1, 4, 29, 48, 58, 87, 89, 106, 149] [2, 5, 49, 50, 59, 90, 105, 107, 167] [6, 32, 51, 61, 83, 88, 90, 91, 152] [3, 7, 33, 52, 62, 89, 92, 110, 153] [4, 8, 34, 53, 63, 90, 93, 111, 154] [5, 9, 54, 55, 64, 94, 110, 112, 169] [10, 36, 55, 65, 84, 91, 93, 95, 156] [6, 11, 37, 56, 66, 92, 94, 96, 157] [7, 12, 38, 57, 67, 93, 97, 115, 158] [8, 13, 39, 58, 68, 94, 98, 116, 159] [9, 14, 59, 60, 69, 95, 99, 115, 117] [10, 15, 41, 60, 70, 96, 98, 100, 161] [11, 16, 42, 61, 71, 97, 99, 101, 162] [12, 17, 43, 62, 72, 98, 102, 120, 163] [13, 18, 44, 63, 73, 85, 99, 103, 121] [14, 19, 64, 65, 74, 100, 104, 120, 122] [15, 20, 46, 65, 75, 101, 103, 105, 165] [16, 21, 47, 66, 76, 102, 104, 106, 166] [17, 22, 48, 67, 77, 86, 103, 107, 125] [18, 23, 49, 68, 78, 87, 104, 108, 126] [0, 19, 24, 69, 70, 105, 109, 125, 127] [20, 25, 51, 70, 79, 106, 108, 110, 168] [21, 26, 52, 71, 80, 88, 107, 109, 111] [22, 27, 53, 72, 81, 89, 108, 112, 130] [1, 23, 28, 54, 73, 90, 109, 113, 131] [2, 24, 29, 74, 75, 110, 114, 130, 132] [25, 30, 56, 75, 82, 91, 111, 113, 115] [26, 31, 57, 76, 83, 92, 112, 114, 116] [3, 27, 32, 58, 77, 93, 113, 117, 135] [4, 28, 33, 59, 78, 94, 114, 118, 136] [0, 5, 29, 34, 79, 115, 119, 135, 137] [30, 35, 61, 79, 84, 96, 116, 118, 120] [6, 31, 36, 62, 80, 97, 117, 119, 121] [7, 32, 37, 63, 81, 98, 118, 122, 140] [1, 8, 33, 38, 64, 99, 119, 123, 141] [2, 9, 34, 39, 82, 120, 124, 140, 142] [10, 35, 40, 66, 82, 101, 121, 123, 125] [11, 36, 41, 67, 83, 102, 122, 124, 126] [3, 12, 37, 42, 68, 103, 123, 127, 145] [4, 13, 38, 43, 69, 104, 124, 128, 146] [5, 14, 39, 44, 84, 125, 129, 145, 147] [15, 40, 45, 71, 84, 106, 126, 128, 130] [6, 16, 41, 46, 72, 107, 127, 129, 131] [7, 17, 42, 47, 73, 108, 128, 132, 150] [8, 18, 43, 48, 74, 109, 129, 133, 151] [9, 10, 19, 44, 49, 130, 134, 150, 152] [10, 20, 45, 50, 76, 111, 131, 133, 135] [11, 21, 46, 51, 77, 112, 132, 134, 136] [12, 22, 47, 52, 78, 113, 133, 137, 155] [0, 13, 23, 48, 53, 114, 134, 138, 156] [14, 15, 24, 49, 54, 135, 139, 155, 157] [15, 25, 50, 55, 80, 116, 136, 138, 140] [16, 26, 51, 56, 81, 117, 137, 139, 141] [1, 17, 27, 52, 57, 118, 138, 142, 160] [2, 18, 28, 53, 58, 119, 139, 143, 161] [19, 20, 29, 54, 59, 140, 144, 160, 162] [20, 30, 55, 60, 83, 121, 141, 143, 145] [3, 21, 31, 56, 61, 122, 142, 144, 146] [4, 22, 32, 57, 62, 123, 143, 147, 164] [5, 23, 33, 58, 63, 124, 144, 148, 165] [24, 25, 34, 59, 64, 145, 149, 164, 166] [6, 25, 35, 60, 65, 126, 146, 148, 150] [7, 26, 36, 61, 66, 127, 147, 149, 151] [8, 27, 37, 62, 67, 128, 148, 152, 167] [9, 28, 38, 63, 68, 129, 149, 153, 168] [29, 30, 39, 64, 69, 88, 150, 154, 167] [11, 30, 40, 65, 70, 131, 151, 153, 155] [12, 31, 41, 66, 71, 132, 152, 154, 156] [13, 32, 42, 67, 72, 133, 153, 157, 169] [14, 33, 43, 68, 73, 91, 134, 154, 158] [34, 35, 44, 69, 74, 92, 155, 159, 169] [16, 35, 45, 70, 75, 136, 156, 158, 160] [17, 36, 46, 71, 76, 137, 157, 159, 161] [18, 37, 47, 72, 77, 95, 138, 158, 162] [19, 38, 48, 73, 78, 96, 139, 159, 163] [21, 40, 50, 75, 79, 141, 161, 163, 164] [22, 41, 51, 76, 80, 85, 142, 162, 165] [23, 42, 52, 77, 81, 100, 143, 163, 166] [26, 45, 55, 79, 82, 86, 146, 165, 167] [27, 46, 56, 80, 83, 87, 147, 166, 168] [31, 50, 60, 82, 84, 89, 151, 168, 169]
H_Z (85 checks, sparse supports)
[0, 1, 18, 80, 85, 87, 109, 119, 138] [1, 3, 22, 82, 86, 89, 113, 123, 142] [2, 4, 23, 83, 87, 90, 114, 124, 143] [3, 6, 26, 69, 88, 92, 117, 127, 146] [4, 7, 27, 84, 89, 93, 118, 128, 147] [5, 6, 8, 28, 90, 94, 119, 129, 148] [6, 10, 30, 73, 91, 96, 121, 131, 150] [7, 11, 31, 74, 92, 97, 122, 132, 151] [8, 10, 12, 32, 93, 98, 123, 133, 152] [9, 11, 13, 33, 94, 99, 124, 134, 153] [0, 10, 14, 77, 95, 100, 125, 134, 135] [11, 15, 35, 78, 96, 101, 126, 136, 155] [0, 12, 16, 36, 97, 102, 127, 137, 156] [13, 15, 17, 37, 98, 103, 128, 138, 157] [14, 16, 18, 38, 99, 104, 129, 139, 158] [2, 15, 19, 81, 100, 105, 130, 139, 140] [1, 16, 20, 40, 101, 106, 131, 141, 160] [2, 17, 21, 41, 102, 107, 132, 142, 161] [18, 20, 22, 42, 103, 108, 133, 143, 162] [19, 21, 23, 43, 104, 109, 134, 144, 163] [3, 5, 20, 24, 105, 110, 135, 144, 145] [4, 21, 25, 45, 106, 111, 136, 146, 164] [5, 22, 26, 46, 107, 112, 137, 147, 165] [23, 25, 27, 47, 108, 113, 138, 148, 166] [24, 26, 28, 48, 86, 109, 114, 139, 149] [7, 9, 25, 29, 110, 115, 140, 149, 150] [8, 26, 30, 50, 111, 116, 141, 151, 167] [9, 27, 31, 51, 112, 117, 142, 152, 168] [28, 30, 32, 52, 88, 113, 118, 143, 153] [29, 31, 33, 53, 89, 114, 119, 144, 154] [12, 14, 30, 34, 115, 120, 145, 154, 155] [13, 31, 35, 55, 116, 121, 146, 156, 169] [14, 32, 36, 56, 91, 117, 122, 147, 157] [33, 35, 37, 57, 92, 118, 123, 148, 158] [34, 36, 38, 58, 93, 119, 124, 149, 159] [17, 19, 35, 39, 120, 125, 150, 159, 160] [18, 36, 40, 60, 95, 121, 126, 151, 161] [19, 37, 41, 61, 96, 122, 127, 152, 162] [38, 40, 42, 62, 97, 123, 128, 153, 163] [39, 41, 43, 63, 85, 98, 124, 129, 154] [22, 24, 40, 44, 85, 125, 130, 155, 164] [23, 41, 45, 65, 100, 126, 131, 156, 165] [24, 42, 46, 66, 101, 127, 132, 157, 166] [43, 45, 47, 67, 86, 102, 128, 133, 158] [44, 46, 48, 68, 87, 103, 129, 134, 159] [27, 29, 45, 49, 87, 130, 135, 160, 167] [28, 46, 50, 70, 105, 131, 136, 161, 168] [29, 47, 51, 71, 88, 106, 132, 137, 162] [48, 50, 52, 72, 89, 107, 133, 138, 163] [49, 51, 53, 73, 85, 90, 108, 134, 139] [32, 34, 50, 54, 90, 135, 140, 164, 169] [33, 51, 55, 75, 91, 110, 136, 141, 165] [34, 52, 56, 76, 92, 111, 137, 142, 166] [53, 55, 57, 77, 86, 93, 112, 138, 143] [54, 56, 58, 78, 87, 94, 113, 139, 144] [37, 39, 55, 59, 94, 95, 140, 145, 167] [38, 56, 60, 79, 96, 115, 141, 146, 168] [39, 57, 61, 80, 88, 97, 116, 142, 147] [58, 60, 62, 81, 89, 98, 117, 143, 148] [1, 59, 61, 63, 90, 99, 118, 144, 149] [42, 44, 60, 64, 99, 100, 145, 150, 169] [43, 61, 65, 82, 91, 101, 120, 146, 151] [44, 62, 66, 83, 92, 102, 121, 147, 152] [3, 63, 65, 67, 93, 103, 122, 148, 153] [4, 64, 66, 68, 94, 104, 123, 149, 154] [47, 49, 65, 69, 95, 104, 105, 150, 155] [48, 66, 70, 84, 96, 106, 125, 151, 156] [6, 49, 67, 71, 97, 107, 126, 152, 157] [7, 68, 70, 72, 98, 108, 127, 153, 158] [8, 69, 71, 73, 99, 109, 128, 154, 159] [52, 54, 70, 74, 100, 109, 110, 155, 160] [10, 53, 71, 75, 101, 111, 130, 156, 161] [11, 54, 72, 76, 102, 112, 131, 157, 162] [12, 73, 75, 77, 103, 113, 132, 158, 163] [13, 74, 76, 78, 85, 104, 114, 133, 159] [0, 57, 59, 75, 105, 114, 115, 160, 164] [15, 58, 76, 79, 106, 116, 135, 161, 165] [16, 59, 77, 80, 107, 117, 136, 162, 166] [17, 78, 79, 81, 86, 108, 118, 137, 163] [2, 62, 64, 79, 110, 119, 120, 164, 167] [20, 63, 80, 82, 111, 121, 140, 165, 168] [21, 64, 81, 83, 88, 112, 122, 141, 166] [5, 67, 69, 82, 115, 124, 125, 167, 169] [25, 68, 83, 84, 91, 116, 126, 145, 168] [9, 72, 74, 84, 95, 120, 129, 130, 169]
Code ID 170-16-17 · download JSON · raw on GitHub