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[[144,2,13]] d ≤
n
144
k
2
d
13
kd²/n
2.347
w
5
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 13, d_Z ≤ 13 · w_X = 5, w_Z = 5 (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)
[24, 32, 55, 80, 88, 96, 99, 109, 113, 116, 124, 132, 140]
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)
[12, 16, 22, 31, 35, 37, 45, 49, 64, 65, 97, 99, 113]
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 5 · H_Z 5
qubit degrees H_X 2–3 (mean 2.5) · H_Z 2–3 (mean 2.5)
trapping sets H_X (1,2)×72 (2,2)×72 (3,2)×72 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 72 (1,3): 72 (2,2): 72 (2,3): 432 (2,4): 216 (3,2): 72 (3,3): 1320 (3,4): 2376 (3,5): 1008 (3,6): 432 (3,7): 72
trapping sets H_Z (1,2)×72 (2,2)×72 (3,2)×72 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 72 (1,3): 72 (2,2): 72 (2,3): 432 (2,4): 216 (3,2): 72 (3,3): 1320 (3,4): 2376 (3,5): 1008 (3,6): 432 (3,7): 72

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C3 x (C3 : C8) = SmallGroup(72, 12) (order 72), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 144; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 6, 28}, b = {0, 19} as element indices. Check weight 5. 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 ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[144,2,13]] — two-block group-algebra code over C3 x (C3 : C8)

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| = 144

Parameters are also spelled out in provenance.construction of codes/144-2-13.json.

Parity checks

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