[[84,36,6]] · QEC Challenge

[[84,36,6]] d ≤

kd²/n

15.429

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 6, d_Z ≤ 6 · w_X = 18, w_Z = 18 (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 12–18 (mean 12.5) · H_Z 12–18 (mean 12.5)

qubit degrees H_X 2–4 (mean 3.571) · H_Z 2–4 (mean 3.571)

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

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

(1,2): 7 (1,3): 22 (1,4): 55 (2,1): 5 (2,2): 31 (2,3): 91 (2,4): 297 (2,5): 463 (2,6): 629 (3,1): 5 (3,2): 104 (3,3): 607 (3,4): 2055 (3,5): 4793 (3,6): 8999 (3,7): 9044 (3,8): 8250 (3,9): 968 (3,10): 657

trapping sets H_Z (1,2)×8 (2,1)×9 (3,1)×11 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 8 (1,3): 20 (1,4): 56 (2,1): 9 (2,2): 30 (2,3): 83 (2,4): 312 (2,5): 435 (2,6): 653 (3,1): 11 (3,2): 107 (3,3): 562 (3,4): 2162 (3,5): 4726 (3,6): 9144 (3,7): 8629 (3,8): 8788 (3,9): 954 (3,10): 695

Construction & provenance

authors @hirobumii

provenance submitted through the challenge

novelty novelty not audited

construction Two explicitly recorded coefficient hyperplanes in the known Okada–Kasai pair-partition CPM84/34/6 check rowspaces. The parent exponent arrays and literal lift convention are preserved.

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 18, maximum qubit degree 8, 48 checks and total incidence 600.

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

[[84,36,6]] - CSS sparse stabilizer code

Direction & hypothesis

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

What was searched

Two explicitly recorded coefficient hyperplanes in the known Okada–Kasai pair-partition CPM84/34/6 check rowspaces. The parent exponent arrays and literal lift convention are preserved.

The displayed presentation has maximum check weight 18, maximum qubit degree 8, 48 checks and total incidence 600.

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/84-36-6.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,"Z":8970}.

Selected sector coefficient masks: {"X":{"functional":12312,"selected_coefficients":[1,2,4,32,64,128,256,512,1024,2048,16384,32768,65536,131072,262144,524288,1048576,2097152,4194304,8388608,16777216,33030271,8208,524160]},"Z":{"functional":8970,"selected_coefficients":[1,4,16,32,64,128,1024,2048,4096,8191,16384,32768,65536,131072,262144,524288,1048576,2097152,4194304,8388608,16777216,33030271,514,8200]}}.

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 24 (max weight 18) · Z-checks 24 (max weight 18)

H_X (24 checks, sparse supports)

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

H_Z (24 checks, sparse supports)

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

Code ID 84-36-6 · download JSON · raw on GitHub