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[[162,84,2]] d =
n
162
k
84
d
2
kd²/n
2.074
w
9
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · 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 2 · witness weight 2 (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, 2 qubits)
[5, 15]
d_Z 2 · witness weight 2 (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, 2 qubits)
[41, 73]
certificate exact, d = 2 · CryptoMiniSat 5.14.7 SAT
X: no logical < 2 exists; Z: no logical < 2 exists

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 4 · H_Z 4 (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 3–6 (mean 4.5) · H_Z 3–6 (mean 4.5)
trapping sets H_X (1,3)×81 (2,0)×162 (3,0)×243 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 81 (1,6): 81 (2,0): 162 (2,3): 162 (2,7): 972 (2,10): 729 (3,0): 243 (3,3): 1161 (3,6): 2619 (3,7): 1458 (3,8): 7290 (3,9): 2916 (3,11): 14580 (3,13): 729 (3,14): 10935
trapping sets H_Z (1,3)×81 (2,0)×81 (3,3)×837 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 81 (1,6): 81 (2,0): 81 (2,3): 162 (2,6): 405 (2,7): 972 (3,3): 837 (3,4): 1458 (3,5): 972 (3,6): 3159 (3,7): 2916 (3,8): 5103 (3,9): 2835 (3,11): 4860 (3,12): 540

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over (C9 x C3) : C3 = SmallGroup(81, 3) (order 81), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 162; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 2, 8, 9, 20, 38}, b = {0, 2, 9} 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 2bga-nonabelian (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

[[162,84,2]] — two-block group-algebra code over (C9 x C3) : C3

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| = 162

Parameters are also spelled out in provenance.construction of codes/162-84-2.json.

Parity checks

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