[[84,40,6]] · QEC Challenge

[[84,40,6]] d ≤

kd²/n

17.143

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 6, d_Z ≤ 6 · w_X = 20, w_Z = 20 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[0, 2, 13, 28, 36, 76]

d_Z 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[0, 18, 27, 37, 68, 72]

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 12–20 (mean 14.545) · H_Z 12–20 (mean 14.545)

qubit degrees H_X 2–6 (mean 3.81) · H_Z 1–6 (mean 3.81)

trapping sets H_X (1,2)×8 (2,1)×5 (3,1)×9 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 8 (1,3): 27 (1,4): 27 (1,5): 17 (1,6): 5 (2,1): 5 (2,2): 26 (2,3): 120 (2,4): 267 (2,5): 452 (2,6): 459 (2,7): 317 (2,8): 116 (2,9): 33 (2,10): 4 (3,1): 9 (3,2): 133 (3,3): 732 (3,4): 2469 (3,5): 5409 (3,6): 8779 (3,7): 10464 (3,8): 9016 (3,9): 5758 (3,10): 2891 (3,11): 1003 (3,12): 233 (3,13): 36 (3,14): 4

trapping sets H_Z (1,1)×1 (2,1)×5 (3,1)×7 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 1 (1,2): 6 (1,3): 20 (1,4): 39 (1,5): 17 (1,6): 1 (2,1): 5 (2,2): 29 (2,3): 97 (2,4): 272 (2,5): 456 (2,6): 556 (2,7): 353 (2,8): 64 (2,9): 2 (3,1): 7 (3,2): 140 (3,3): 701 (3,4): 2353 (3,5): 5484 (3,6): 9330 (3,7): 11379 (3,8): 10034 (3,9): 5899 (3,10): 2150 (3,11): 627 (3,12): 125 (3,13): 8

Construction & provenance

authors @hirobumii

provenance submitted through the challenge

novelty novelty not audited

construction Known Okada-Kasai pair-partition CPM parent, literal codimension-three CSS subgroups in each sector

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

builds on arXiv:2607.14091

date 2026-10-18

notes Displayed presentation: maximum check weight 20, maximum qubit degree 10, 44 checks and total incidence 640.

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

[[84,40,6]] - CSS sparse stabilizer code

Direction & hypothesis

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

What was searched

Known Okada-Kasai pair-partition CPM parent, literal codimension-three CSS subgroups in each sector

The displayed presentation has maximum check weight 20, maximum qubit degree 10, 44 checks and total incidence 640.

Evidence trail

The JSON contains weight-6 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/84-40-6.json. Allocate zero binary matrices H_X and H_Z with 84 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.

Parent-basis parity functionals: {"X":[12312,826,34242],"Z":[8970,346,18455]}.

Selected sector coefficient masks: {"X":{"functionals":[12312,826,34242],"selected_coefficients":[1,4,2048,16384,65536,131072,262144,524288,1048576,2097152,4194304,8388608,16777216,33030271,544,1088,32832,524160,192,14568859,20083545,27056563]},"Z":{"functionals":[8970,346,18455],"selected_coefficients":[32,128,1024,4096,32768,65536,131072,262144,524288,1048576,2097152,4194304,8388608,16777216,33030271,264,18432,2052,8175,8704,16385,27053529]}}.

CPM lift parameters: {"p":7,"J":4,"L":12,"EX":[[0,4,0,4,5,6,2,4,5,0,4,6],[5,1,6,3,5,1,3,1,6,3,0,3],[3,0,5,0,2,6,3,1,6,6,3,6],[4,3,2,4,1,4,2,1,6,1,2,1]],"EZ":[[4,0,1,6,5,5,6,0,6,2,4,5],[1,6,0,4,2,0,2,5,0,4,5,6],[1,3,5,0,6,1,5,5,6,6,6,4],[4,4,5,6,5,2,2,2,2,3,6,6]]}.

Public source: 84-34-6, contributed by Koki Okada and Kenta Kasai.

Construction references:

arXiv:2607.14091

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

Parity checks

X-checks 22 (max weight 20) · Z-checks 22 (max weight 20)

H_X (22 checks, sparse supports)

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

H_Z (22 checks, sparse supports)

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

Code ID 84-40-6 · download JSON · raw on GitHub