[[80,45,5]] · QEC Challenge

[[80,45,5]] d ≤

kd²/n

14.062

X/Z

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Distance

X/Z asymmetry 1 · d_X ≤ 5, d_Z ≤ 5 · w_X = 20, w_Z = 19 (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)

[10, 16, 29, 31, 38]

d_Z 5 · witness weight 5 (claimed upper_bound)

witness operator (support, 5 qubits)

[4, 9, 21, 23, 39]

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 14–20 (mean 18.765) · H_Z 12–19 (mean 16.167)

qubit degrees H_X 1–8 (mean 3.987) · H_Z 1–7 (mean 3.638)

trapping sets H_X (1,1)×2 (2,1)×10 (3,1)×53 (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): 8 (1,3): 21 (1,4): 23 (1,5): 14 (1,6): 9 (1,7): 2 (1,8): 1 (2,1): 10 (2,2): 44 (2,3): 146 (2,4): 288 (2,5): 434 (2,6): 440 (2,7): 326 (2,8): 185 (2,9): 85 (2,10): 22 (2,11): 11 (2,13): 1 (3,1): 53 (3,2): 349 (3,3): 1329 (3,4): 3685 (3,5): 7133 (3,6): 10147 (3,7): 11013 (3,8): 9156 (3,9): 6062 (3,10): 3026 (3,11): 1272 (3,12): 435 (3,13): 91 (3,14): 17 (3,15): 3

trapping sets H_Z (1,1)×3 (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_Z

(1,1): 3 (1,2): 11 (1,3): 27 (1,4): 19 (1,5): 14 (1,6): 3 (1,7): 3 (2,1): 9 (2,2): 50 (2,3): 171 (2,4): 315 (2,5): 389 (2,6): 365 (2,7): 232 (2,8): 118 (2,9): 46 (2,10): 11 (2,11): 2 (2,12): 1 (3,1): 62 (3,2): 358 (3,3): 1359 (3,4): 3634 (3,5): 6535 (3,6): 8785 (3,7): 8827 (3,8): 6842 (3,9): 4037 (3,10): 1959 (3,11): 734 (3,12): 230 (3,13): 54 (3,14): 4

Construction & provenance

authors @hirobumii

provenance submitted through the challenge

novelty novelty not audited

construction Take the explicitly recorded coefficient hyperplane(s) in the accepted80/44/5 parent, then apply the recorded ordered physical CNOT sequence.

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 20, maximum qubit degree 12, 35 checks and total incidence 610.

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

[[80,45,5]] - CSS sparse stabilizer code

Direction & hypothesis

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

What was searched

Take the explicitly recorded coefficient hyperplane(s) in the 80/44/5 parent, then apply the recorded ordered physical CNOT sequence.

The displayed presentation has maximum check weight 20, maximum qubit degree 12, 35 checks and total incidence 610.

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/80-45-5.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: [[23,64],[35,65],[37,44],[48,33],[49,26],[61,77],[72,13],[22,1],[22,78],[52,53],[15,67],[22,11]].

Parent-basis parity functionals: {"X":20279,"Z":null}.

Source sector rows as integer support masks (bit q is qubit q): {"X":["12102506812355261760010","83831805625108203323394","765028058516213328057344","10081145710431662342168","45051705255894123088008","47242833274631977961857","37939201463722137151760","377789544379141347889356","114572728578233200551956","161870274983536432840768","162481335810094050324514","180206396550827542119944","323934049330023165331042","113670002626300085936197","457248672104818386434179","761060959228182877745223","1095688499556747046843650"],"Z":["81461686520672921092352","26600206292566624174179","95037823661497022189572","297503859532191981636","1317481304082220320","11144247490658739945493","89946396163219126782468","359273981413080821792768","4796276182265075870744","38079845360929375556004","96822641324253252682368","646983017455779909871617","935083905251118701875200","56809054306440013218440","41995169816876970672144","151150640702114590998536","316445754609636399392050","605422226643688827652163"]}. 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 17 (max weight 20) · Z-checks 18 (max weight 19)

H_X (17 checks, sparse supports)

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

H_Z (18 checks, sparse supports)

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

Code ID 80-45-5 · download JSON · raw on GitHub