[[80,32,6]] · QEC Challenge

[[80,32,6]] d ≤

kd²/n

14.4

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 6, d_Z ≤ 6 · w_X = 10, w_Z = 10 (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)

[11, 15, 18, 27, 51, 70]

d_Z 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[7, 33, 37, 46, 63, 78]

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 10 · H_Z 10

qubit degrees H_X 2–4 (mean 3.0) · H_Z 2–4 (mean 3.0)

trapping sets H_X (1,2)×24 (2,1)×8 (3,1)×24 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 24 (1,3): 32 (1,4): 24 (2,1): 8 (2,2): 36 (2,3): 216 (2,4): 332 (2,5): 304 (2,6): 112 (3,1): 24 (3,2): 144 (3,3): 1048 (3,4): 2920 (3,5): 4776 (3,6): 4472 (3,7): 3032 (3,8): 1144 (3,9): 280 (3,10): 32

trapping sets H_Z (1,2)×24 (2,1)×8 (3,1)×24 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

Construction & provenance

authors @hirobumii

provenance submitted through the challenge

novelty novelty not audited

construction Rank-24-per-sector CSS subgroup of the public [[80,24,8]] 2BGA code from PR #3773 (https://github.com/unitaryfoundation/qldpc-challenge/pull/3773). Select original parent rows by zero-based bit masks X=877032374220 and Z=879599236095.

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

date 2026-10-18

notes Model attribution records the multi-agent configuration documented on 2026-10-17: GPT-6 Astra for construction, adversarial tests and key review, and GPT-6.1 Sol for literature research. This is workflow-configuration attribution; a candidate-specific runtime record was not preserved.

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

[[80,32,6]] - rank-24 CSS subgroup of a public 2BGA code

Direction & hypothesis

Increase the encoded dimension of the public [[80,24,8]] 2BGA code by retaining a smaller stabilizer subgroup, while keeping maximum check weight 10. The parent was contributed by @FarLab in PR #3773.

What was searched

The selected subgroup keeps 24 original X rows and 24 original Z rows. Their zero-based row masks are X = 877032374220 and Z = 879599236095. Both selected sector ranks are 24, giving k = 32. The displayed basis has maximum check weight 10, maximum qubit degree 7, 48 checks and total incidence 480.

Evidence trail

The submission tool searches for logical witnesses and verifies the generated JSON. Its sector distance claims are witness-backed upper bounds. The retained exact-distance research check excludes all nonzero centralizer words through weight 5: 3,241 syndromes for errors through weight 2 are distinct, and all 82,160 triples avoid that set. Genuine weight-6 quotient witnesses establish exact distance 6 separately.

Dead ends

The fixed-parent degree-6 census leaves 100 admissible binary selections, all with sector ranks 20 and 20. None spans the required 24 plus 24 ranks; the selected degree-7 presentation avoids that obstruction.

Tools

Recorded workflow configuration: GPT-6 Astra + GPT-6.1 Sol. The official qldpc submit run used 20,000 RIS trials per sector with seed 0, followed by its independent refutation check.

Reproduction

Take the parent's 80-24-8 JSON at fa4b0de3bef2b078ffa41860af8d6ee1633931a8. Its Git blob is 43dae9d893a9f2b5c37480d6df4dbc2e10ac423f. Preserve qubit labels and original row order. Keep X row i when bit i of 877032374220 is 1, and Z row i when bit i of 879599236095 is 1, starting at bit 0. This directly reproduces the submitted matrices.

Parity checks

X-checks 24 (max weight 10) · Z-checks 24 (max weight 10)

H_X (24 checks, sparse supports)

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

H_Z (24 checks, sparse supports)

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

Code ID 80-32-6 · download JSON · raw on GitHub