[[78,4,12]] · QEC Challenge

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[[78,4,12]] d ≤

kd²/n

7.385

X/Z

0.0064

5.831

layers

swaps

434

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Distance

X/Z asymmetry 1 · d_X ≤ 12, d_Z ≤ 12 · 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 12 · witness weight 12 (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, 12 qubits)

[5, 17, 22, 31, 32, 36, 46, 51, 58, 63, 67, 72]

d_Z 12 · witness weight 12 (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, 12 qubits)

[14, 22, 31, 32, 39, 47, 50, 56, 64, 72, 73, 76]

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 9 · H_Z 9

qubit degrees H_X 4–5 (mean 4.5) · H_Z 4–5 (mean 4.5)

trapping sets H_X (1,4)×39 (2,4)×39 (3,4)×78 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 39 (1,5): 39 (2,4): 39 (2,5): 117 (2,6): 234 (2,7): 546 (2,8): 273 (3,4): 78 (3,5): 351 (3,6): 1404 (3,7): 2379 (3,8): 2574 (3,9): 7280 (3,10): 7098 (3,11): 2457 (3,12): 390 (3,13): 117

trapping sets H_Z (1,4)×39 (2,4)×39 (3,4)×78 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 39 (1,5): 39 (2,4): 39 (2,5): 117 (2,6): 234 (2,7): 546 (2,8): 273 (3,4): 78 (3,5): 351 (3,6): 1404 (3,7): 2379 (3,8): 2574 (3,9): 7280 (3,10): 7098 (3,11): 2457 (3,12): 390 (3,13): 117

witness diameter X 5.831 · Z 6.0828 (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 (40)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 434 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 C13 : C3 = SmallGroup(39, 1) (order 39), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 78; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 1, 2, 3, 8}, b = {0, 2, 23, 29} 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

[[78,4,12]] — two-block group-algebra code over C13 : C3

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| = 78

Parameters are also spelled out in provenance.construction of codes/78-4-12.json.

Parity checks

X-checks 39 (max weight 9) · Z-checks 39 (max weight 9)

H_X (39 checks, sparse supports)

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

H_Z (39 checks, sparse supports)

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

Code ID 78-4-12 · download JSON · raw on GitHub