[[88,2,14]] · QEC Challenge

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[[88,2,14]] d ≤

kd²/n

4.455

X/Z

0.0034

6.0

layers

swaps

471

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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)

[3, 28, 31, 32, 33, 40, 45, 48, 60, 63, 72, 74, 79, 87]

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)

[10, 12, 23, 25, 33, 41, 48, 49, 52, 65, 70, 72, 73, 82]

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)×88 (2,4)×154 (3,4)×440 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,4): 88 (2,4): 154 (2,6): 924 (3,4): 440 (3,6): 3476 (3,8): 13728 (3,10): 924

trapping sets H_Z (1,4)×88 (2,4)×154 (3,4)×440 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,4): 88 (2,4): 154 (2,6): 924 (3,4): 440 (3,6): 3476 (3,8): 13728 (3,10): 924

witness diameter X 6.7082 · Z 6.3246 (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 (46)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 471 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 D44 = SmallGroup(44, 3) (order 44), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 88; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 1, 3, 26}, b = {0, 6, 11, 31} 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

[[88,2,14]] — two-block group-algebra code over D44

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| = 88

Parameters are also spelled out in provenance.construction of codes/88-2-14.json.

Parity checks

X-checks 44 (max weight 8) · Z-checks 44 (max weight 8)

H_X (44 checks, sparse supports)

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

H_Z (44 checks, sparse supports)

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

Code ID 88-2-14 · download JSON · raw on GitHub