[[72,4,11]] · QEC Challenge

[[72,4,11]] d ≤

kd²/n

6.722

X/Z

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Distance

X/Z asymmetry 1 · d_X ≤ 11, d_Z ≤ 11 · w_X = 7, w_Z = 7 (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, 28, 31, 32, 45, 47, 48, 49, 55, 65, 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)

[2, 8, 20, 22, 29, 31, 35, 49, 54, 57, 71]

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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 7 · H_Z 7

qubit degrees H_X 3–4 (mean 3.5) · H_Z 3–4 (mean 3.5)

trapping sets H_X (1,3)×36 (2,4)×108 (3,3)×36 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 36 (1,4): 36 (2,4): 108 (2,5): 432 (2,6): 216 (3,3): 36 (3,4): 144 (3,5): 756 (3,6): 3420 (3,7): 4356 (3,8): 1836 (3,9): 648 (3,10): 144

trapping sets H_Z (1,3)×36 (2,4)×108 (3,3)×36 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 36 (1,4): 36 (2,4): 108 (2,5): 432 (2,6): 216 (3,3): 36 (3,4): 144 (3,5): 756 (3,6): 3420 (3,7): 4356 (3,8): 1836 (3,9): 648 (3,10): 144

Construction & provenance

authors Lee, Gideon

provenance literature baseline

construction Two-block group-algebra code C(a,b) over C36 = SmallGroup(36, 2) (order 36), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 72; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 1, 9, 25}, b = {0, 10, 13} as element indices. Check weight 7. 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 unrestricted (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

[[72,4,11]] — two-block group-algebra code over C36

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| = 72

Parameters are also spelled out in provenance.construction of codes/72-4-11.json.

Parity checks

X-checks 36 (max weight 7) · Z-checks 36 (max weight 7)

H_X (36 checks, sparse supports)

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

H_Z (36 checks, sparse supports)

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

Code ID 72-4-11 · download JSON · raw on GitHub