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[[126,54,3]] d =
n
126
k
54
d
3
kd²/n
3.857
w
9
X/Z
1
g
0.0119
r
4.2426
layers
2
swaps
162

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Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · 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 3 · witness weight 3 (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, 3 qubits)
[15, 57, 78]
d_Z 3 · witness weight 3 (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, 3 qubits)
[22, 103, 105]
certificate exact, d = 3 · CryptoMiniSat 5.14.7 SAT
X: no logical < 3 exists; Z: no logical < 3 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)×63 (2,3)×126 (3,0)×63 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 63 (1,6): 63 (2,3): 126 (2,4): 189 (2,6): 63 (2,7): 756 (2,8): 189 (2,10): 378 (3,0): 63 (3,3): 378 (3,4): 756 (3,6): 2583 (3,7): 3843 (3,8): 2268 (3,9): 2709 (3,10): 3402 (3,11): 4914 (3,13): 378 (3,14): 2709
trapping sets H_Z (1,3)×63 (2,3)×126 (3,0)×63 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 63 (1,6): 63 (2,3): 126 (2,4): 189 (2,6): 63 (2,7): 756 (2,8): 189 (2,10): 378 (3,0): 63 (3,3): 378 (3,4): 756 (3,6): 2583 (3,7): 3843 (3,8): 2268 (3,9): 2709 (3,10): 3402 (3,11): 4914 (3,13): 378 (3,14): 2709
witness diameter X 2.0 · Z 2.2361 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
layout contributed by @MathysRennela · Simulated annealing over integer-grid site assignments with the repo's own research/local2d/fold_layout.search_layout, single layer at radius 4 then two layers at 7, plus a divisor scan of the two-block index convention Z_l x Z_m which gives the tighter embedding here; radius and per-site occupancy re-measured with the verifier's own rule. · 2026-10-13
r = 4.243
X checkZ checkqubit site (63)2 qubits stacked (2 layers)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release
routing cost 162 nearest-neighbor SWAPs per round in total, at most 5 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C7 : C9 = SmallGroup(63, 1) (order 63), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 126; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 3, 4, 11, 35, 47}, b = {0, 3, 35} 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 2D-local bilayer (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

[[126,54,3]] — two-block group-algebra code over C7 : C9

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| = 126

Parameters are also spelled out in provenance.construction of codes/126-54-3.json.

Parity checks

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