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[[96,4,14]] d ≤
n
96
k
4
d
14
kd²/n
8.167
w
8
X/Z
1
g
0.0063
r
6.0
layers
2
swaps
475

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Distance

X/Z asymmetry 1 · d_X ≤ 14, d_Z ≤ 14 · 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 14 · witness weight 14 (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, 14 qubits)
[11, 27, 37, 39, 49, 54, 57, 60, 61, 74, 78, 81, 89, 93]
d_Z 14 · witness weight 14 (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, 14 qubits)
[3, 9, 16, 22, 25, 28, 36, 38, 53, 58, 65, 76, 81, 84]
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 4 · H_Z 4 (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)×96 (2,4)×240 (3,4)×288 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 96 (2,4): 240 (2,6): 864 (3,4): 288 (3,6): 5664 (3,8): 10992 (3,10): 768
trapping sets H_Z (1,4)×96 (2,4)×240 (3,4)×288 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 96 (2,4): 240 (2,6): 864 (3,4): 288 (3,6): 5664 (3,8): 10992 (3,10): 768
witness diameter X 7.2111 · Z 6.4031 (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 = 6
X checkZ checkqubit site (49)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 475 nearest-neighbor SWAPs per round in total, at most 10 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 D48 = SmallGroup(48, 7) (order 48), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 96; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 1, 2, 21}, b = {0, 11, 14, 41} 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 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

[[96,4,14]] — two-block group-algebra code over D48

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| = 96

Parameters are also spelled out in provenance.construction of codes/96-4-14.json.

Parity checks

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