[[93,29,7]] · QEC Challenge

[[93,29,7]] d ≤

kd²/n

15.28

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 7, d_Z ≤ 7 · w_X = 12, w_Z = 12 (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)

[7, 14, 30, 37, 49, 52, 53]

d_Z 7 · witness weight 7 (claimed upper_bound)

witness operator (support, 7 qubits)

[2, 8, 32, 38, 40, 52, 56]

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 7–12 (mean 8.735) · H_Z 8–12 (mean 11.267)

qubit degrees H_X 1–5 (mean 3.194) · H_Z 2–6 (mean 3.634)

trapping sets H_X (1,1)×2 (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

(1,1): 2 (1,2): 21 (1,3): 34 (1,4): 29 (1,5): 7 (2,1): 5 (2,2): 34 (2,3): 141 (2,4): 267 (2,5): 324 (2,6): 225 (2,7): 87 (2,8): 11 (3,1): 5 (3,2): 89 (3,3): 508 (3,4): 1591 (3,5): 3279 (3,6): 4479 (3,7): 4361 (3,8): 2960 (3,9): 1397 (3,10): 447 (3,11): 102 (3,12): 24 (3,13): 1

trapping sets H_Z (1,2)×4 (2,1)×1 (3,1)×5 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 4 (1,3): 40 (1,4): 37 (1,5): 10 (1,6): 2 (2,1): 1 (2,2): 15 (2,3): 59 (2,4): 285 (2,5): 549 (2,6): 426 (2,7): 192 (2,8): 48 (2,9): 13 (2,10): 1 (3,1): 5 (3,2): 52 (3,3): 311 (3,4): 1264 (3,5): 3830 (3,6): 7876 (3,7): 9552 (3,8): 7486 (3,9): 4268 (3,10): 1790 (3,11): 606 (3,12): 115 (3,13): 32 (3,14): 1

Construction & provenance

authors @hirobumii

provenance submitted through the challenge

novelty novelty not audited

construction Apply physical CNOT(control79,target43), zero-based, to the source93/29/7 I636 CSS stabilizer group, then use the explicit minimum-incidence basis from its complete source stabilizer pools through weight13. The source has polynomial-minor seed a=37,b=268566560,c=1140850720 over F2[x]/(x³¹+1), followed by the recorded Koszul, rank-hyperplane and physical-kernel changes. Its candidate SHA256 is 7253bc420e645a12e92fb230a8f8a9b0cca82534c202b91e23d28b5d93a004b3. Select X source cap13 pool indices [0, 1, 2, 9, 3, 4, 5, 6, 7, 8, 10, 14, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 36, 39] and Z source cap13 pool indices [0, 1, 2, 3, 4, 5, 6, 7, 27, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29]; transform X target43 by control79 and Z control79 by target43. The supplied literal matrices have ranks34/30 and resource vector(93,29,7,12,9,64,635).

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

builds on https://arxiv.org/abs/quant-ph/0304125, https://arxiv.org/abs/2601.18879

date 2026-10-17

notes Displayed presentation: maximum check weight 12, maximum qubit degree 9, 64 checks and total incidence 635.

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

[[93,29,7]] - CSS sparse stabilizer code

Direction & hypothesis

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

What was searched

The source has polynomial-minor seed a=37,b=268566560,c=1140850720 over F2[x]/(x^31+1), followed by the recorded Koszul, rank-hyperplane and physical-kernel changes. Its candidate SHA256 is 7253bc420e645a12e92fb230a8f8a9b0cca82534c202b91e23d28b5d93a004b3. Select X source cap13 pool indices [0, 1, 2, 9, 3, 4, 5, 6, 7, 8, 10, 14, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 36, 39] and Z source cap13 pool indices [0, 1, 2, 3, 4, 5, 6, 7, 27, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29]; transform X target43 by control79 and Z control79 by target43. The supplied literal matrices have ranks34/30 and resource vector(93,29,7,12,9,64,635).

The displayed presentation has maximum check weight 12, maximum qubit degree 9, 64 checks and total incidence 635.

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/93-29-7.json. Allocate zero binary matrices H_X and H_Z with 93 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.

Physical operation: "CNOT(control=79,target=43) after choosing the literal source pool bases below".

Construction references:

https://arxiv.org/abs/quant-ph/0304125
https://arxiv.org/abs/2601.18879

Local validation: the official verify/validate_candidate.py passed, including independent-seed distance refutation.

Parity checks

X-checks 34 (max weight 12) · Z-checks 30 (max weight 12)

H_X (34 checks, sparse supports)

[6, 19, 48, 50, 54, 65, 67] [16, 29, 33, 58, 60, 75, 77] [1, 8, 13, 36, 52, 69, 82] [31, 37, 59, 71, 76, 79, 87] [17, 24, 28, 43, 45, 64, 79, 91] [13, 18, 21, 29, 35, 42, 44, 48] [7, 22, 27, 30, 44, 51, 53, 57] [1, 9, 24, 29, 46, 53, 55, 59] [36, 42, 58, 61, 70, 75, 78, 86] [0, 6, 22, 25, 63, 65, 69, 87] [32, 38, 44, 60, 72, 77, 80, 88] [5, 15, 23, 34, 59, 76, 79, 80] [0, 3, 5, 31, 39, 50, 62, 68, 72] [2, 5, 7, 33, 41, 52, 64, 70, 74] [4, 7, 9, 35, 43, 54, 66, 72, 76] [5, 9, 30, 31, 34, 36, 62, 70, 81] [10, 13, 15, 41, 49, 60, 72, 78, 82] [12, 18, 22, 44, 47, 49, 63, 75, 83] [12, 15, 17, 31, 43, 51, 74, 80, 84] [3, 9, 13, 35, 38, 40, 66, 74, 85] [14, 17, 19, 33, 45, 53, 76, 82, 86] [0, 26, 29, 34, 45, 57, 63, 67, 88] [16, 19, 21, 35, 47, 55, 78, 84, 88] [22, 28, 30, 46, 74, 83, 86, 87, 89] [8, 14, 20, 25, 52, 54, 64, 85, 90] [18, 21, 23, 37, 49, 57, 80, 86, 90] [14, 16, 18, 43, 51, 62, 66, 68, 91] [7, 9, 10, 40, 44, 54, 56, 70, 92] [2, 4, 30, 38, 49, 61, 67, 71, 92] [10, 16, 20, 42, 45, 47, 73, 81, 92] [20, 23, 25, 39, 51, 59, 82, 88, 92] [6, 8, 35, 51, 56, 64, 67, 83, 86, 89] [0, 4, 11, 12, 27, 28, 34, 41, 62, 68, 69, 75] [4, 6, 11, 27, 40, 41, 56, 59, 71, 73, 76, 84]

H_Z (30 checks, sparse supports)

[19, 21, 32, 48, 58, 77, 86, 89] [1, 8, 21, 24, 35, 66, 90, 91] [1, 5, 14, 37, 41, 59, 62, 82, 90] [10, 34, 43, 62, 71, 72, 79, 84, 88, 92] [0, 23, 25, 28, 38, 50, 64, 67, 74, 80] [11, 16, 27, 44, 47, 48, 54, 66, 77, 85] [12, 17, 19, 28, 37, 49, 61, 65, 78, 87] [14, 18, 42, 47, 49, 61, 78, 83, 86, 90] [16, 20, 22, 29, 35, 43, 46, 51, 63, 85, 91] [3, 5, 12, 30, 34, 49, 57, 60, 62, 68, 74, 77] [0, 4, 6, 13, 35, 50, 58, 61, 63, 69, 75, 78] [1, 5, 7, 14, 31, 36, 51, 59, 64, 70, 76, 79] [2, 6, 8, 15, 32, 37, 52, 60, 65, 71, 77, 80] [3, 7, 9, 16, 33, 38, 53, 61, 66, 72, 78, 81] [4, 8, 10, 17, 31, 34, 39, 54, 67, 73, 79, 82] [5, 9, 11, 18, 32, 35, 40, 55, 68, 74, 80, 83] [6, 10, 12, 19, 33, 36, 41, 56, 69, 75, 81, 84] [7, 11, 13, 20, 34, 37, 42, 57, 70, 76, 82, 85] [0, 18, 22, 24, 37, 45, 48, 53, 62, 65, 81, 87] [9, 13, 15, 22, 36, 39, 44, 59, 72, 78, 84, 87] [0, 7, 25, 29, 44, 52, 55, 60, 63, 69, 72, 88] [10, 14, 16, 23, 37, 40, 45, 60, 73, 79, 85, 88] [1, 8, 26, 30, 45, 53, 56, 61, 64, 70, 73, 89] [2, 20, 24, 26, 39, 47, 50, 55, 64, 67, 83, 89] [11, 15, 17, 24, 38, 41, 46, 61, 74, 80, 86, 89] [0, 2, 9, 27, 31, 46, 54, 57, 65, 71, 74, 90] [3, 21, 25, 27, 40, 48, 51, 56, 65, 68, 84, 90] [4, 22, 26, 28, 41, 49, 52, 57, 66, 69, 85, 91] [2, 4, 11, 29, 33, 48, 56, 59, 67, 73, 76, 92] [17, 21, 23, 30, 36, 44, 47, 52, 64, 80, 86, 92]

Code ID 93-29-7 · download JSON · raw on GitHub