← back to the board
[[124,10,13]] d ≤
n
124
k
10
d
13
kd²/n
13.629
w
7
X/Z
1

Share this result

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)
[4, 5, 9, 11, 14, 46, 51, 60, 64, 71, 76, 105, 121]
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)
[6, 13, 14, 29, 59, 71, 74, 80, 92, 97, 98, 100, 106]
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)×62 (2,4)×186 (3,3)×62 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 62 (1,4): 62 (2,4): 186 (2,5): 744 (2,6): 372 (3,3): 62 (3,4): 124 (3,5): 682 (3,6): 6200 (3,7): 9362 (3,8): 3348 (3,9): 1116 (3,10): 248
trapping sets H_Z (1,3)×62 (2,4)×186 (3,3)×62 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 62 (1,4): 62 (2,4): 186 (2,5): 744 (2,6): 372 (3,3): 62 (3,4): 124 (3,5): 682 (3,6): 6200 (3,7): 9362 (3,8): 3348 (3,9): 1116 (3,10): 248

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, 3, 41, 46}, b = {0, 5, 11} 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

[[124,10,13]] — 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/124-10-13.json.

Parity checks

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