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[[124,14,13]] d ≤
n
124
k
14
d
13
kd²/n
19.081
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 13, d_Z ≤ 13 · 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 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)
[25, 26, 30, 37, 38, 60, 63, 66, 89, 104, 119, 122, 123]
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)
[18, 23, 26, 28, 31, 41, 49, 69, 70, 74, 78, 92, 109]
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)×124 (2,6)×1736 (3,6)×1612 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 124 (2,6): 1736 (3,6): 1612 (3,8): 31620 (3,10): 3472
trapping sets H_Z (1,4)×124 (2,6)×1736 (3,6)×1612 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 124 (2,6): 1736 (3,6): 1612 (3,8): 31620 (3,10): 3472

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C62 = SmallGroup(62, 2) (order 62), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 124; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 2, 5, 23}, b = {0, 4, 35, 57} 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

[[124,14,13]] — two-block group-algebra code over C62

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| = 124

Parameters are also spelled out in provenance.construction of codes/124-14-13.json.

Parity checks

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