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[[80,41,5]] d ≤
n
80
k
41
d
5
kd²/n
12.812
w
15
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 5, d_Z ≤ 5 · w_X = 15, w_Z = 15 (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)
[11, 12, 13, 22, 26]
d_Z 5 · witness weight 5 (claimed upper_bound)
witness operator (support, 5 qubits)
[2, 11, 14, 19, 20]
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 10–15 (mean 13.55) · H_Z 10–15 (mean 13.789)
qubit degrees H_X 1–6 (mean 3.388) · H_Z 1–6 (mean 3.275)
trapping sets H_X (1,1)×2 (2,1)×9 (3,1)×62 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 2 (1,2): 16 (1,3): 28 (1,4): 21 (1,5): 9 (1,6): 4 (2,1): 9 (2,2): 65 (2,3): 172 (2,4): 331 (2,5): 335 (2,6): 250 (2,7): 166 (2,8): 61 (2,9): 13 (2,10): 1 (3,1): 62 (3,2): 321 (3,3): 1309 (3,4): 3178 (3,5): 5573 (3,6): 6715 (3,7): 6242 (3,8): 4589 (3,9): 2410 (3,10): 982 (3,11): 313 (3,12): 64 (3,13): 10
trapping sets H_Z (1,1)×1 (2,1)×14 (3,1)×87 (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): 20 (1,3): 30 (1,4): 17 (1,5): 9 (1,6): 3 (2,1): 14 (2,2): 63 (2,3): 227 (2,4): 361 (2,5): 338 (2,6): 228 (2,7): 122 (2,8): 42 (2,9): 15 (3,1): 87 (3,2): 433 (3,3): 1676 (3,4): 4058 (3,5): 6310 (3,6): 6939 (3,7): 5798 (3,8): 3749 (3,9): 1893 (3,10): 758 (3,11): 203 (3,12): 49 (3,13): 7 (3,14): 1

Construction & provenance

authors @hirobumii
provenance submitted through the challenge
novelty novelty not audited
construction Apply nine recorded ordered CNOTs to the literal80/41/5 hyperplane source, then rebase within the same transformed CSS stabilizer group.
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 15, maximum qubit degree 11, 39 checks and total incidence 533.
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

[[80,41,5]] - CSS sparse stabilizer code

Direction & hypothesis

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

What was searched

Apply nine recorded ordered CNOTs to the literal80/41/5 hyperplane source, then rebase within the same transformed CSS stabilizer group.

The displayed presentation has maximum check weight 15, maximum qubit degree 11, 39 checks and total incidence 533.

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/80-41-5.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: [[7,2],[9,0],[20,14],[24,11],[32,69],[48,42],[58,71],[65,3],[68,6]].

Source sector rows as integer support masks (bit q is qubit q): {"X":["21398439863983959704073","151779905948787620708384","306990863497869771538566","5904120067509786248224","9463260783409432167488","453515544994616629604354","738590338911314215045","47224818877201602347264","75557940852302599573511","191408032984370239399944","311687722714766838343696","609187005673546296328386","319364499387259052032","311971625412404491321920","756464081678369206063168","10081154714332381347856","38977972508685752338448","79146332630385842978888","305856567786215707771137","377844671266561313482764"],"Z":["56742476396467358532112","81461686942851026419968","4142447617586155753472","9446050306305884292352","95037823661492727218176","302527183812771001278496","311796884339445010007314","623371155754747598996488","2352299576189345888","77334520286257762795664","3283522627792616292418","11070460375825436115077","38079845079450104004868","151152659370382164541568","306990789184072538528289","628097873836348432590880","66872051323334099592","80427876221223669106688","907284960021705237201474"]}. Apply the CNOTs above in order: X[target] ^= X[control], Z[control] ^= Z[target], then choose the displayed basis within the resulting row spaces.

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