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[[80,26,7]] d ≤
n
80
k
26
d
7
kd²/n
15.925
w
10
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 7, d_Z ≤ 7 · 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 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[0, 1, 3, 37, 49, 51, 79]
d_Z 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[0, 6, 12, 30, 64, 66, 74]
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 1–5 (mean 3.375) · H_Z 1–5 (mean 3.375)
trapping sets H_X (1,1)×3 (2,1)×3 (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
(1,1): 3 (1,2): 6 (1,3): 40 (1,4): 20 (1,5): 11 (2,1): 3 (2,2): 18 (2,3): 81 (2,4): 303 (2,5): 304 (2,6): 273 (2,7): 101 (2,8): 28 (3,1): 5 (3,2): 66 (3,3): 399 (3,4): 1316 (3,5): 3434 (3,6): 4816 (3,7): 5267 (3,8): 3631 (3,9): 1992 (3,10): 589 (3,11): 165 (3,12): 34 (3,13): 3
trapping sets H_Z (1,1)×2 (2,1)×2 (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): 2 (1,2): 15 (1,3): 25 (1,4): 27 (1,5): 11 (2,1): 2 (2,2): 25 (2,3): 105 (2,4): 215 (2,5): 323 (2,6): 277 (2,7): 133 (2,8): 28 (3,1): 7 (3,2): 72 (3,3): 401 (3,4): 1317 (3,5): 3092 (3,6): 4550 (3,7): 5084 (3,8): 4035 (3,9): 2247 (3,10): 835 (3,11): 199 (3,12): 37 (3,13): 2

Construction & provenance

authors @hirobumii
provenance submitted through the challenge
novelty novelty not audited
construction Rank-one-per-sector stabilizer subgroup of the public @FarLab 2BGA code [[80,24,8]] on ab(5,2,2,2). Parent exact Git blob43dae9d893a9f2b5c37480d6df4dbc2e10ac423f, JSON SHA256a35525e02f719c21c60a548078777d280286010cd0ef5f5eb5fae93db27653a6, at https://github.com/unitaryfoundation/qldpc-challenge/blob/15e58eea2bba862eb49a8a162fa801f81d027f5a/codes/80-24-8.json. Select these zero-indexed rows directly from its literal checks: X=[24, 32, 5, 16, 25, 4, 9, 8, 27, 35, 36, 28, 21, 15, 0, 20, 29, 3, 38, 13, 30, 23, 2, 39, 12, 22, 31];Z=[22, 32, 7, 27, 23, 6, 33, 9, 24, 10, 34, 4, 35, 11, 21, 3, 30, 19, 2, 13, 31, 14, 1, 28, 0, 39, 17]. The resulting ranks are27+27 and actual(n,k,d,w,D,M,I)=[80, 26, 7, 10, 8, 54, 540]. No entangling transformation or inherited distance assumption is used.
model GPT-6 Astra + GPT-6.1 Sol (recorded multi-agent configuration) (claimed, not verified)
builds on https://github.com/unitaryfoundation/qldpc-challenge/blob/15e58eea2bba862eb49a8a162fa801f81d027f5a/codes/80-24-8.json
date 2026-10-17
notes Displayed presentation: maximum check weight 10, maximum qubit degree 8, 54 checks and total incidence 540.
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

[[80,26,7]] - CSS sparse stabilizer code

Direction & hypothesis

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

What was searched

Rank-one-per-sector stabilizer subgroup of the public @FarLab 2BGA code [[80,24,8]] on ab(5,2,2,2). Parent exact Git blob43dae9d893a9f2b5c37480d6df4dbc2e10ac423f, JSON SHA256a35525e02f719c21c60a548078777d280286010cd0ef5f5eb5fae93db27653a6, at https://github.com/unitaryfoundation/qldpc-challenge/blob/15e58eea2bba862eb49a8a162fa801f81d027f5a/codes/80-24-8.json. Select these zero-indexed rows directly from its literal checks: X=[24, 32, 5, 16, 25, 4, 9, 8, 27, 35, 36, 28, 21, 15, 0, 20, 29, 3, 38, 13, 30, 23, 2, 39, 12, 22, 31];Z=[22, 32, 7, 27, 23, 6, 33, 9, 24, 10, 34, 4, 35, 11, 21, 3, 30, 19, 2, 13, 31, 14, 1, 28, 0, 39, 17]. The resulting ranks are27+27 and actual(n,k,d,w,D,M,I)=[80, 26, 7, 10, 8, 54, 540].

The displayed presentation has maximum check weight 10, maximum qubit degree 8, 54 checks and total incidence 540.

Evidence trail

The JSON contains weight-7 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/80-26-7.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.

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 27 (max weight 10) · Z-checks 27 (max weight 10)
H_X (27 checks, sparse supports)
[0, 14, 18, 24, 39, 41, 51, 61, 64, 72] [7, 8, 22, 26, 32, 40, 49, 59, 69, 72] [5, 10, 21, 27, 39, 45, 53, 60, 70, 72] [6, 10, 16, 31, 32, 43, 53, 56, 64, 73] [1, 15, 19, 25, 38, 40, 50, 60, 65, 73] [4, 11, 20, 26, 38, 44, 52, 61, 71, 73] [3, 9, 22, 25, 39, 44, 49, 57, 64, 74] [2, 8, 23, 24, 38, 45, 48, 56, 65, 75] [3, 13, 17, 27, 36, 42, 48, 62, 67, 75] [4, 11, 21, 25, 35, 43, 50, 56, 70, 75] [3, 12, 18, 30, 36, 44, 53, 63, 65, 76] [4, 10, 22, 28, 35, 45, 55, 57, 68, 76] [3, 15, 21, 26, 37, 46, 48, 61, 69, 76] [5, 15, 16, 31, 33, 42, 55, 63, 70, 76] [0, 15, 16, 30, 34, 40, 48, 57, 67, 77] [2, 14, 20, 27, 36, 47, 49, 60, 68, 77] [5, 11, 23, 29, 34, 44, 54, 56, 69, 77] [3, 12, 19, 29, 33, 43, 51, 58, 64, 78] [1, 14, 16, 28, 38, 46, 55, 61, 67, 78] [7, 13, 18, 29, 35, 40, 53, 61, 68, 78] [6, 8, 20, 30, 33, 47, 53, 59, 70, 78] [1, 13, 23, 24, 39, 44, 50, 63, 71, 78] [2, 13, 18, 28, 32, 42, 50, 59, 65, 79] [0, 15, 17, 29, 39, 47, 54, 60, 66, 79] [6, 12, 19, 28, 34, 41, 52, 60, 69, 79] [0, 12, 22, 25, 38, 45, 51, 62, 70, 79] [7, 9, 21, 31, 32, 46, 52, 58, 71, 79]
H_Z (27 checks, sparse supports)
[7, 14, 22, 27, 37, 46, 49, 62, 68, 72] [5, 11, 17, 24, 32, 42, 54, 56, 71, 72] [7, 10, 20, 30, 39, 47, 53, 57, 71, 72] [0, 10, 19, 27, 38, 45, 51, 60, 67, 73] [6, 15, 23, 26, 36, 47, 48, 63, 69, 73] [6, 11, 21, 31, 38, 46, 52, 56, 70, 73] [4, 10, 16, 25, 33, 43, 55, 57, 70, 73] [1, 9, 20, 26, 32, 46, 49, 59, 71, 73] [3, 9, 16, 24, 37, 46, 48, 63, 64, 74] [2, 10, 23, 25, 35, 45, 50, 56, 68, 74] [7, 9, 19, 26, 34, 40, 52, 58, 69, 74] [4, 9, 23, 29, 36, 44, 54, 58, 68, 75] [6, 8, 18, 27, 35, 41, 53, 59, 68, 75] [3, 11, 22, 24, 34, 44, 51, 57, 69, 75] [4, 13, 21, 24, 38, 45, 50, 61, 71, 75] [3, 14, 16, 26, 35, 43, 49, 61, 67, 76] [5, 15, 22, 30, 35, 40, 54, 57, 70, 76] [2, 11, 19, 30, 32, 43, 52, 59, 65, 77] [2, 15, 17, 27, 34, 42, 48, 60, 66, 77] [5, 13, 16, 30, 36, 42, 53, 63, 67, 77] [4, 14, 23, 31, 34, 41, 55, 56, 71, 77] [6, 14, 19, 29, 39, 41, 54, 60, 64, 78] [1, 12, 18, 24, 33, 41, 51, 63, 65, 78] [7, 13, 20, 28, 33, 42, 52, 59, 68, 78] [0, 13, 19, 25, 32, 40, 50, 62, 64, 79] [2, 12, 22, 31, 39, 45, 49, 63, 64, 79] [0, 9, 17, 28, 34, 41, 54, 57, 67, 79]
Code ID 80-26-7 · download JSON · raw on GitHub