[[70,6,11]] · QEC Challenge

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[[70,6,11]] d ≤

kd²/n

10.371

X/Z

0.009

5.831

layers

swaps

392

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Distance

X/Z asymmetry 1 · d_X ≤ 11, d_Z ≤ 11 · 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 11 · witness weight 11 (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, 11 qubits)

[0, 4, 20, 23, 27, 30, 34, 37, 61, 63, 68]

d_Z 11 · witness weight 11 (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, 11 qubits)

[3, 6, 7, 10, 18, 28, 30, 31, 39, 53, 58]

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)×35 (2,4)×35 (3,4)×70 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 35 (1,5): 35 (2,4): 35 (2,5): 70 (2,6): 175 (2,7): 560 (2,8): 280 (3,4): 70 (3,5): 140 (3,6): 630 (3,7): 2100 (3,8): 3605 (3,9): 6790 (3,10): 6510 (3,11): 2765 (3,12): 665 (3,13): 175

trapping sets H_Z (1,4)×35 (2,4)×35 (3,4)×70 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 35 (1,5): 35 (2,4): 35 (2,5): 70 (2,6): 175 (2,7): 560 (2,8): 280 (3,4): 70 (3,5): 140 (3,6): 630 (3,7): 2100 (3,8): 3605 (3,9): 6790 (3,10): 6510 (3,11): 2765 (3,12): 665 (3,13): 175

witness diameter X 6.4031 · Z 5.6569 (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 (36)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 392 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 C35 = SmallGroup(35, 1) (order 35), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 70; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 1, 2, 3, 23}, b = {0, 2, 14, 16} 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 generalized bicycle (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

[[70,6,11]] — two-block group-algebra code over C35

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| = 70

Parameters are also spelled out in provenance.construction of codes/70-6-11.json.

Parity checks

X-checks 35 (max weight 9) · Z-checks 35 (max weight 9)

H_X (35 checks, sparse supports)

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

H_Z (35 checks, sparse supports)

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

Code ID 70-6-11 · download JSON · raw on GitHub