[[96,24,6]] · QEC Challenge

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[[96,24,6]] d =

kd²/n

9.0

X/Z

0.0054

6.4031

layers

swaps

634

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Distance

X/Z asymmetry 1 · d_X = 6, d_Z = 6 · 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 6 · witness weight 6 (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, 6 qubits)

[19, 26, 39, 50, 84, 90]

d_Z 6 · witness weight 6 (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, 6 qubits)

[7, 27, 42, 58, 82, 88]

certificate exact, d = 6 · CryptoMiniSat 5.14.7 SAT

X: no logical < 6 exists; Z: no logical < 6 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 8 · H_Z 8

qubit degrees H_X 4 · H_Z 4

trapping sets H_X (1,4)×96 (2,4)×24 (3,6)×2112 (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): 24 (2,6): 1296 (3,6): 2112 (3,8): 22224 (3,10): 2400

trapping sets H_Z (1,4)×96 (2,4)×24 (3,6)×2112 (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): 24 (2,6): 1296 (3,6): 2112 (3,8): 22224 (3,10): 2400

witness diameter X 6.0828 · 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

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 634 nearest-neighbor SWAPs per round in total, at most 11 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 C6 x Q8 = SmallGroup(48, 46) (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, 11, 13}, b = {0, 2, 8, 29} 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

[[96,24,6]] — two-block group-algebra code over C6 x Q8

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/96-24-6.json.

Parity checks

X-checks 48 (max weight 8) · Z-checks 48 (max weight 8)

H_X (48 checks, sparse supports)

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

H_Z (48 checks, sparse supports)

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

Code ID 96-24-6 · download JSON · raw on GitHub