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[[170,16,13]] d ≤
n
170
k
16
d
13
kd²/n
15.906
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 13, d_Z ≤ 13 · w_X = 7, w_Z = 7 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 13 · witness weight 13 (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, 13 qubits)
[1, 8, 9, 53, 60, 61, 62, 72, 74, 81, 145, 146, 166]
d_Z 13 · witness weight 13 (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, 13 qubits)
[0, 37, 57, 88, 89, 90, 122, 132, 141, 150, 151, 154, 163]
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 7 · H_Z 7
qubit degrees H_X 3–4 (mean 3.5) · H_Z 3–4 (mean 3.5)
trapping sets H_X (1,3)×85 (2,4)×255 (3,3)×85 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 85 (1,4): 85 (2,4): 255 (2,5): 1020 (2,6): 510 (3,3): 85 (3,4): 85 (3,5): 1020 (3,6): 8755 (3,7): 12580 (3,8): 4590 (3,9): 1530 (3,10): 340
trapping sets H_Z (1,3)×85 (2,4)×255 (3,3)×85 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 85 (1,4): 85 (2,4): 255 (2,5): 1020 (2,6): 510 (3,3): 85 (3,4): 85 (3,5): 1020 (3,6): 8755 (3,7): 12580 (3,8): 4590 (3,9): 1530 (3,10): 340

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, 11, 46, 76}, b = {0, 4, 5} as element indices. Check weight 7. 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

[[170,16,13]] — two-block group-algebra code over C85

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| = 170

Parameters are also spelled out in provenance.construction of codes/170-16-13.json.

Parity checks

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