[[80,36,5]] · QEC Challenge

[[80,36,5]] d ≤

kd²/n

11.25

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 5, d_Z ≤ 5 · w_X = 13, w_Z = 13 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 5 · witness weight 5 (claimed upper_bound)

witness operator (support, 5 qubits)

[15, 16, 27, 30, 38]

d_Z 5 · witness weight 5 (claimed upper_bound)

witness operator (support, 5 qubits)

[0, 4, 15, 24, 51]

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 8–13 (mean 11.273) · H_Z 8–13 (mean 10.818)

qubit degrees H_X 1–5 (mean 3.1) · H_Z 1–5 (mean 2.975)

trapping sets H_X (1,1)×4 (2,1)×11 (3,1)×51 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 4 (1,2): 20 (1,3): 30 (1,4): 16 (1,5): 10 (2,1): 11 (2,2): 71 (2,3): 171 (2,4): 313 (2,5): 278 (2,6): 197 (2,7): 70 (2,8): 22 (3,1): 51 (3,2): 343 (3,3): 1204 (3,4): 2829 (3,5): 4539 (3,6): 5043 (3,7): 4203 (3,8): 2463 (3,9): 1102 (3,10): 371 (3,11): 79 (3,12): 13 (3,13): 1

trapping sets H_Z (1,1)×4 (2,1)×12 (3,1)×61 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 4 (1,2): 23 (1,3): 31 (1,4): 15 (1,5): 7 (2,1): 12 (2,2): 79 (2,3): 200 (2,4): 315 (2,5): 251 (2,6): 151 (2,7): 45 (2,8): 10 (3,1): 61 (3,2): 397 (3,3): 1358 (3,4): 3057 (3,5): 4463 (3,6): 4488 (3,7): 3372 (3,8): 1730 (3,9): 686 (3,10): 202 (3,11): 34 (3,12): 3

Construction & provenance

authors @hirobumii

provenance submitted through the challenge

novelty novelty not audited

construction A rank22-per-sector hyperplane subgroup of the exact80/34/5 CSS image, followed by the specified ordered physical CNOT sequence. Parent coefficient functional269572 in each sector.

model GPT-6 Astra + GPT-6.1 Sol (recorded multi-agent configuration) (claimed, not verified)

date 2026-10-18

notes Displayed presentation: maximum check weight 13, maximum qubit degree 8, 44 checks and total incidence 486.

family other (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

[[80,36,5]] - CSS sparse stabilizer code

Direction & hypothesis

Study the dimension–distance tradeoff obtained by selecting stabilizer subgroups and changing sparse presentations.

What was searched

A rank22-per-sector hyperplane subgroup of the 80/34/5 CSS image, followed by the specified ordered physical CNOT sequence. Parent coefficient functional269572 in each sector.

The displayed presentation has maximum check weight 13, maximum qubit degree 8, 44 checks and total incidence 486.

Evidence trail

The JSON contains weight-5 logical witnesses. Distance confidence: X: upper_bound, Z: upper_bound.

Tools

Recorded workflow configuration: GPT-6 Astra + GPT-6.1 Sol. Official tooling: verify/qldpc_verify.py.

Reproduction

Read codes/80-36-5.json. Allocate zero binary matrices H_X and H_Z with 80 columns. For each row list in checks.X or checks.Z, set the listed zero-based columns to 1, preserving row order. This reconstructs the displayed presentation exactly.

Ordered CNOT pairs (control,target), zero-based: [[9,41],[19,60],[23,50],[28,34],[35,2],[37,20],[40,30],[51,61],[59,8],[63,1],[66,29],[67,14],[69,52],[71,11],[73,26],[74,22],[79,18],[54,39],[59,49],[0,72],[29,66],[40,26],[60,2],[67,77],[74,67]].

Parent-basis parity functionals: {"X":269572,"Z":269572}.

Public source: 80-24-8, contributed by @FarLab.

Local validation: the official verify/validate_candidate.py passed, including independent-seed distance refutation.

Parity checks

X-checks 22 (max weight 13) · Z-checks 22 (max weight 13)

H_X (22 checks, sparse supports)

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

H_Z (22 checks, sparse supports)

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

Code ID 80-36-5 · download JSON · raw on GitHub