[[108,8,12]] · QEC Challenge

[[108,8,12]] d ≤

108

kd²/n

10.667

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 12, d_Z ≤ 12 · w_X = 26, w_Z = 26 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 12 · witness weight 12 (claimed upper_bound)

witness operator (support, 12 qubits)

[65, 68, 71, 74, 77, 80, 92, 95, 98, 101, 104, 107]

d_Z 12 · witness weight 12 (claimed upper_bound)

witness operator (support, 12 qubits)

[11, 14, 17, 20, 23, 26, 38, 41, 44, 47, 50, 53]

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 26 · H_Z 26

qubit degrees H_X 8–18 (mean 13.0) · H_Z 8–18 (mean 13.0)

trapping sets H_X (1,8)×54 (2,8)×27 (3,10)×270 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,8): 54 (1,18): 54 (2,8): 27 (2,10): 54 (2,12): 324 (2,14): 648 (2,16): 216 (2,18): 432 (2,20): 918 (2,22): 1296 (2,24): 783 (2,26): 216 (2,28): 270 (2,30): 54 (3,10): 270 (3,12): 666 (3,14): 2862 (3,16): 5724 (3,18): 17172 (3,20): 23814 (3,22): 33102 (3,24): 39636 (3,26): 37152 (3,28): 21114 (3,30): 9990 (3,32): 3294 (3,34): 1350 (3,36): 378 (3,38): 108

trapping sets H_Z (1,8)×54 (2,8)×27 (3,10)×270 (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 @mathysrennela

provenance submitted through the challenge

novelty novelty not audited

construction Bivariate bicycle on Z_6 x Z_9; A has 13 terms and B has 13 terms; exact supports are in the submitted JSON.

model GPT-5.6 Luna (claimed, not verified)

date 2026-08-18

family bivariate bicycle (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

[[108,8,12]] — bivariate-bicycle weight-26 candidate

Direction & hypothesis

Target the unrestricted high-weight frontier at raw weight 26, filling a gap between the bounded-weight tracks and the existing weight-30 high-rate constructions.

What was searched

A seeded random sweep of 1,200 bivariate-bicycle instances on moderate tori, with raw check weights 25 through 29. Rank survivors were screened with 1,500-trial RIS. The submitted instance is on Z_6 x Z_9, with 13 A terms and 13 B terms, giving maximum check weight 26.

Evidence trail

The submission CLI reran both CSS directions with 8,000 RIS trials and obtained d_X<=12 and d_Z<=12. The submitted claim is d<=12. The full local verifier passed the CSS, rank, weight, witness, and fresh refutation checks. This is not an exact-distance claim.

Dead ends

A generalized-bicycle Z_333 mutation route retained high k only at weight 30; reducing its support total to 25–29 collapsed the rank to at most 6. Screen claims that collapsed under fresh RIS were corrected or excluded.

Tools

Model: GPT-5.6 Luna. Author: @mathysrennela. The search used the repository bivariate-bicycle constructor, GF(2) rank, RIS screening, and the trusted verifier.

Reproduction

Rebuild the exact checks from codes/108-8-12.json; its X/Z support lists are the complete reproducible artifact. The construction is a bivariate bicycle on Z_6 x Z_9 with the listed supports. The final witness search used 8,000 RIS trials.

Parity checks

X-checks 54 (max weight 26) · Z-checks 54 (max weight 26)

H_X (54 checks, sparse supports)

[0, 3, 6, 9, 13, 18, 19, 24, 32, 33, 34, 38, 39, 45, 46, 48, 49, 50, 54, 55, 62, 67, 82, 85, 89, 93] [1, 4, 7, 10, 14, 19, 20, 25, 33, 34, 35, 39, 40, 46, 47, 49, 50, 51, 54, 55, 56, 68, 81, 83, 86, 94] [2, 5, 8, 11, 15, 20, 21, 26, 27, 34, 35, 40, 41, 47, 48, 50, 51, 52, 55, 56, 57, 69, 82, 84, 87, 95] [0, 3, 6, 12, 16, 18, 21, 22, 27, 28, 35, 41, 42, 48, 49, 51, 52, 53, 56, 57, 58, 70, 83, 85, 88, 96] [1, 4, 7, 13, 17, 19, 22, 23, 27, 28, 29, 42, 43, 45, 49, 50, 52, 53, 57, 58, 59, 71, 84, 86, 89, 97] [2, 5, 8, 9, 14, 20, 23, 24, 28, 29, 30, 43, 44, 45, 46, 50, 51, 53, 58, 59, 60, 63, 81, 85, 87, 98] [0, 3, 6, 10, 15, 21, 24, 25, 29, 30, 31, 36, 44, 45, 46, 47, 51, 52, 59, 60, 61, 64, 82, 86, 88, 90] [1, 4, 7, 11, 16, 22, 25, 26, 30, 31, 32, 36, 37, 46, 47, 48, 52, 53, 60, 61, 62, 65, 83, 87, 89, 91] [2, 5, 8, 12, 17, 18, 23, 26, 31, 32, 33, 37, 38, 45, 47, 48, 49, 53, 54, 61, 62, 66, 81, 84, 88, 92] [0, 1, 3, 4, 5, 9, 12, 15, 18, 22, 27, 28, 33, 41, 42, 43, 47, 48, 63, 64, 71, 76, 91, 94, 98, 102] [1, 2, 4, 5, 6, 10, 13, 16, 19, 23, 28, 29, 34, 42, 43, 44, 48, 49, 63, 64, 65, 77, 90, 92, 95, 103] [2, 3, 5, 6, 7, 11, 14, 17, 20, 24, 29, 30, 35, 36, 43, 44, 49, 50, 64, 65, 66, 78, 91, 93, 96, 104] [3, 4, 6, 7, 8, 9, 12, 15, 21, 25, 27, 30, 31, 36, 37, 44, 50, 51, 65, 66, 67, 79, 92, 94, 97, 105] [0, 4, 5, 7, 8, 10, 13, 16, 22, 26, 28, 31, 32, 36, 37, 38, 51, 52, 66, 67, 68, 80, 93, 95, 98, 106] [0, 1, 5, 6, 8, 11, 14, 17, 18, 23, 29, 32, 33, 37, 38, 39, 52, 53, 67, 68, 69, 72, 90, 94, 96, 107] [0, 1, 2, 6, 7, 9, 12, 15, 19, 24, 30, 33, 34, 38, 39, 40, 45, 53, 68, 69, 70, 73, 91, 95, 97, 99] [1, 2, 3, 7, 8, 10, 13, 16, 20, 25, 31, 34, 35, 39, 40, 41, 45, 46, 69, 70, 71, 74, 92, 96, 98, 100] [0, 2, 3, 4, 8, 11, 14, 17, 21, 26, 27, 32, 35, 40, 41, 42, 46, 47, 63, 70, 71, 75, 90, 93, 97, 101] [2, 3, 9, 10, 12, 13, 14, 18, 21, 24, 27, 31, 36, 37, 42, 50, 51, 52, 57, 72, 73, 80, 85, 100, 103, 107] [3, 4, 10, 11, 13, 14, 15, 19, 22, 25, 28, 32, 37, 38, 43, 51, 52, 53, 58, 72, 73, 74, 86, 99, 101, 104] [4, 5, 11, 12, 14, 15, 16, 20, 23, 26, 29, 33, 38, 39, 44, 45, 52, 53, 59, 73, 74, 75, 87, 100, 102, 105] [5, 6, 12, 13, 15, 16, 17, 18, 21, 24, 30, 34, 36, 39, 40, 45, 46, 53, 60, 74, 75, 76, 88, 101, 103, 106] [6, 7, 9, 13, 14, 16, 17, 19, 22, 25, 31, 35, 37, 40, 41, 45, 46, 47, 61, 75, 76, 77, 89, 102, 104, 107] [7, 8, 9, 10, 14, 15, 17, 20, 23, 26, 27, 32, 38, 41, 42, 46, 47, 48, 62, 76, 77, 78, 81, 99, 103, 105] [0, 8, 9, 10, 11, 15, 16, 18, 21, 24, 28, 33, 39, 42, 43, 47, 48, 49, 54, 77, 78, 79, 82, 100, 104, 106] [0, 1, 10, 11, 12, 16, 17, 19, 22, 25, 29, 34, 40, 43, 44, 48, 49, 50, 55, 78, 79, 80, 83, 101, 105, 107] [1, 2, 9, 11, 12, 13, 17, 20, 23, 26, 30, 35, 36, 41, 44, 49, 50, 51, 56, 72, 79, 80, 84, 99, 102, 106] [5, 6, 7, 11, 12, 18, 19, 21, 22, 23, 27, 30, 33, 36, 40, 45, 46, 51, 55, 58, 62, 66, 81, 82, 89, 94] [6, 7, 8, 12, 13, 19, 20, 22, 23, 24, 28, 31, 34, 37, 41, 46, 47, 52, 54, 56, 59, 67, 81, 82, 83, 95] [0, 7, 8, 13, 14, 20, 21, 23, 24, 25, 29, 32, 35, 38, 42, 47, 48, 53, 55, 57, 60, 68, 82, 83, 84, 96] [0, 1, 8, 14, 15, 21, 22, 24, 25, 26, 27, 30, 33, 39, 43, 45, 48, 49, 56, 58, 61, 69, 83, 84, 85, 97] [0, 1, 2, 15, 16, 18, 22, 23, 25, 26, 28, 31, 34, 40, 44, 46, 49, 50, 57, 59, 62, 70, 84, 85, 86, 98] [1, 2, 3, 16, 17, 18, 19, 23, 24, 26, 29, 32, 35, 36, 41, 47, 50, 51, 54, 58, 60, 71, 85, 86, 87, 90] [2, 3, 4, 9, 17, 18, 19, 20, 24, 25, 27, 30, 33, 37, 42, 48, 51, 52, 55, 59, 61, 63, 86, 87, 88, 91] [3, 4, 5, 9, 10, 19, 20, 21, 25, 26, 28, 31, 34, 38, 43, 49, 52, 53, 56, 60, 62, 64, 87, 88, 89, 92] [4, 5, 6, 10, 11, 18, 20, 21, 22, 26, 29, 32, 35, 39, 44, 45, 50, 53, 54, 57, 61, 65, 81, 88, 89, 93] [0, 1, 6, 14, 15, 16, 20, 21, 27, 28, 30, 31, 32, 36, 39, 42, 45, 49, 64, 67, 71, 75, 90, 91, 98, 103] [1, 2, 7, 15, 16, 17, 21, 22, 28, 29, 31, 32, 33, 37, 40, 43, 46, 50, 63, 65, 68, 76, 90, 91, 92, 104] [2, 3, 8, 9, 16, 17, 22, 23, 29, 30, 32, 33, 34, 38, 41, 44, 47, 51, 64, 66, 69, 77, 91, 92, 93, 105] [0, 3, 4, 9, 10, 17, 23, 24, 30, 31, 33, 34, 35, 36, 39, 42, 48, 52, 65, 67, 70, 78, 92, 93, 94, 106] [1, 4, 5, 9, 10, 11, 24, 25, 27, 31, 32, 34, 35, 37, 40, 43, 49, 53, 66, 68, 71, 79, 93, 94, 95, 107] [2, 5, 6, 10, 11, 12, 25, 26, 27, 28, 32, 33, 35, 38, 41, 44, 45, 50, 63, 67, 69, 80, 94, 95, 96, 99] [3, 6, 7, 11, 12, 13, 18, 26, 27, 28, 29, 33, 34, 36, 39, 42, 46, 51, 64, 68, 70, 72, 95, 96, 97, 100] [4, 7, 8, 12, 13, 14, 18, 19, 28, 29, 30, 34, 35, 37, 40, 43, 47, 52, 65, 69, 71, 73, 96, 97, 98, 101] [0, 5, 8, 13, 14, 15, 19, 20, 27, 29, 30, 31, 35, 38, 41, 44, 48, 53, 63, 66, 70, 74, 90, 97, 98, 102] [0, 4, 9, 10, 15, 23, 24, 25, 29, 30, 36, 37, 39, 40, 41, 45, 48, 51, 58, 73, 76, 80, 84, 99, 100, 107] [1, 5, 10, 11, 16, 24, 25, 26, 30, 31, 37, 38, 40, 41, 42, 46, 49, 52, 59, 72, 74, 77, 85, 99, 100, 101] [2, 6, 11, 12, 17, 18, 25, 26, 31, 32, 38, 39, 41, 42, 43, 47, 50, 53, 60, 73, 75, 78, 86, 100, 101, 102] [3, 7, 9, 12, 13, 18, 19, 26, 32, 33, 39, 40, 42, 43, 44, 45, 48, 51, 61, 74, 76, 79, 87, 101, 102, 103] [4, 8, 10, 13, 14, 18, 19, 20, 33, 34, 36, 40, 41, 43, 44, 46, 49, 52, 62, 75, 77, 80, 88, 102, 103, 104] [0, 5, 11, 14, 15, 19, 20, 21, 34, 35, 36, 37, 41, 42, 44, 47, 50, 53, 54, 72, 76, 78, 89, 103, 104, 105] [1, 6, 12, 15, 16, 20, 21, 22, 27, 35, 36, 37, 38, 42, 43, 45, 48, 51, 55, 73, 77, 79, 81, 104, 105, 106] [2, 7, 13, 16, 17, 21, 22, 23, 27, 28, 37, 38, 39, 43, 44, 46, 49, 52, 56, 74, 78, 80, 82, 105, 106, 107] [3, 8, 9, 14, 17, 22, 23, 24, 28, 29, 36, 38, 39, 40, 44, 47, 50, 53, 57, 72, 75, 79, 83, 99, 106, 107]

H_Z (54 checks, sparse supports)

[0, 1, 8, 24, 28, 32, 35, 50, 54, 57, 60, 63, 67, 68, 69, 71, 78, 79, 83, 84, 85, 90, 93, 98, 99, 104] [0, 1, 2, 25, 27, 29, 33, 51, 55, 58, 61, 63, 64, 68, 69, 70, 79, 80, 84, 85, 86, 90, 91, 94, 100, 105] [1, 2, 3, 26, 28, 30, 34, 52, 56, 59, 62, 64, 65, 69, 70, 71, 72, 80, 85, 86, 87, 91, 92, 95, 101, 106] [2, 3, 4, 18, 29, 31, 35, 53, 54, 57, 60, 63, 65, 66, 70, 71, 72, 73, 86, 87, 88, 92, 93, 96, 102, 107] [3, 4, 5, 19, 27, 30, 32, 45, 55, 58, 61, 63, 64, 66, 67, 71, 73, 74, 87, 88, 89, 93, 94, 97, 99, 103] [4, 5, 6, 20, 28, 31, 33, 46, 56, 59, 62, 63, 64, 65, 67, 68, 74, 75, 81, 88, 89, 94, 95, 98, 100, 104] [5, 6, 7, 21, 29, 32, 34, 47, 54, 57, 60, 64, 65, 66, 68, 69, 75, 76, 81, 82, 89, 90, 95, 96, 101, 105] [6, 7, 8, 22, 30, 33, 35, 48, 55, 58, 61, 65, 66, 67, 69, 70, 76, 77, 81, 82, 83, 91, 96, 97, 102, 106] [0, 7, 8, 23, 27, 31, 34, 49, 56, 59, 62, 66, 67, 68, 70, 71, 77, 78, 82, 83, 84, 92, 97, 98, 103, 107] [5, 9, 10, 17, 33, 37, 41, 44, 54, 59, 63, 66, 69, 72, 76, 77, 78, 80, 87, 88, 92, 93, 94, 99, 102, 107] [6, 9, 10, 11, 34, 36, 38, 42, 55, 60, 64, 67, 70, 72, 73, 77, 78, 79, 88, 89, 93, 94, 95, 99, 100, 103] [7, 10, 11, 12, 35, 37, 39, 43, 56, 61, 65, 68, 71, 73, 74, 78, 79, 80, 81, 89, 94, 95, 96, 100, 101, 104] [8, 11, 12, 13, 27, 38, 40, 44, 57, 62, 63, 66, 69, 72, 74, 75, 79, 80, 81, 82, 95, 96, 97, 101, 102, 105] [0, 12, 13, 14, 28, 36, 39, 41, 54, 58, 64, 67, 70, 72, 73, 75, 76, 80, 82, 83, 96, 97, 98, 102, 103, 106] [1, 13, 14, 15, 29, 37, 40, 42, 55, 59, 65, 68, 71, 72, 73, 74, 76, 77, 83, 84, 90, 97, 98, 103, 104, 107] [2, 14, 15, 16, 30, 38, 41, 43, 56, 60, 63, 66, 69, 73, 74, 75, 77, 78, 84, 85, 90, 91, 98, 99, 104, 105] [3, 15, 16, 17, 31, 39, 42, 44, 57, 61, 64, 67, 70, 74, 75, 76, 78, 79, 85, 86, 90, 91, 92, 100, 105, 106] [4, 9, 16, 17, 32, 36, 40, 43, 58, 62, 65, 68, 71, 75, 76, 77, 79, 80, 86, 87, 91, 92, 93, 101, 106, 107] [14, 18, 19, 26, 42, 46, 50, 53, 54, 57, 62, 63, 68, 72, 75, 78, 81, 85, 86, 87, 89, 96, 97, 101, 102, 103] [15, 18, 19, 20, 43, 45, 47, 51, 54, 55, 58, 64, 69, 73, 76, 79, 81, 82, 86, 87, 88, 97, 98, 102, 103, 104] [16, 19, 20, 21, 44, 46, 48, 52, 55, 56, 59, 65, 70, 74, 77, 80, 82, 83, 87, 88, 89, 90, 98, 103, 104, 105] [17, 20, 21, 22, 36, 47, 49, 53, 56, 57, 60, 66, 71, 72, 75, 78, 81, 83, 84, 88, 89, 90, 91, 104, 105, 106] [9, 21, 22, 23, 37, 45, 48, 50, 57, 58, 61, 63, 67, 73, 76, 79, 81, 82, 84, 85, 89, 91, 92, 105, 106, 107] [10, 22, 23, 24, 38, 46, 49, 51, 58, 59, 62, 64, 68, 74, 77, 80, 81, 82, 83, 85, 86, 92, 93, 99, 106, 107] [11, 23, 24, 25, 39, 47, 50, 52, 54, 59, 60, 65, 69, 72, 75, 78, 82, 83, 84, 86, 87, 93, 94, 99, 100, 107] [12, 24, 25, 26, 40, 48, 51, 53, 55, 60, 61, 66, 70, 73, 76, 79, 83, 84, 85, 87, 88, 94, 95, 99, 100, 101] [13, 18, 25, 26, 41, 45, 49, 52, 56, 61, 62, 67, 71, 74, 77, 80, 84, 85, 86, 88, 89, 95, 96, 100, 101, 102] [1, 5, 8, 23, 27, 28, 35, 51, 56, 57, 58, 63, 66, 71, 72, 77, 81, 84, 87, 90, 94, 95, 96, 98, 105, 106] [0, 2, 6, 24, 27, 28, 29, 52, 57, 58, 59, 63, 64, 67, 73, 78, 82, 85, 88, 90, 91, 95, 96, 97, 106, 107] [1, 3, 7, 25, 28, 29, 30, 53, 58, 59, 60, 64, 65, 68, 74, 79, 83, 86, 89, 91, 92, 96, 97, 98, 99, 107] [2, 4, 8, 26, 29, 30, 31, 45, 59, 60, 61, 65, 66, 69, 75, 80, 81, 84, 87, 90, 92, 93, 97, 98, 99, 100] [0, 3, 5, 18, 30, 31, 32, 46, 60, 61, 62, 66, 67, 70, 72, 76, 82, 85, 88, 90, 91, 93, 94, 98, 100, 101] [1, 4, 6, 19, 31, 32, 33, 47, 54, 61, 62, 67, 68, 71, 73, 77, 83, 86, 89, 90, 91, 92, 94, 95, 101, 102] [2, 5, 7, 20, 32, 33, 34, 48, 54, 55, 62, 63, 68, 69, 74, 78, 81, 84, 87, 91, 92, 93, 95, 96, 102, 103] [3, 6, 8, 21, 33, 34, 35, 49, 54, 55, 56, 64, 69, 70, 75, 79, 82, 85, 88, 92, 93, 94, 96, 97, 103, 104] [0, 4, 7, 22, 27, 34, 35, 50, 55, 56, 57, 65, 70, 71, 76, 80, 83, 86, 89, 93, 94, 95, 97, 98, 104, 105] [6, 10, 14, 17, 32, 36, 37, 44, 60, 61, 65, 66, 67, 72, 75, 80, 81, 86, 90, 93, 96, 99, 103, 104, 105, 107] [7, 9, 11, 15, 33, 36, 37, 38, 61, 62, 66, 67, 68, 72, 73, 76, 82, 87, 91, 94, 97, 99, 100, 104, 105, 106] [8, 10, 12, 16, 34, 37, 38, 39, 54, 62, 67, 68, 69, 73, 74, 77, 83, 88, 92, 95, 98, 100, 101, 105, 106, 107] [0, 11, 13, 17, 35, 38, 39, 40, 54, 55, 68, 69, 70, 74, 75, 78, 84, 89, 90, 93, 96, 99, 101, 102, 106, 107] [1, 9, 12, 14, 27, 39, 40, 41, 55, 56, 69, 70, 71, 75, 76, 79, 81, 85, 91, 94, 97, 99, 100, 102, 103, 107] [2, 10, 13, 15, 28, 40, 41, 42, 56, 57, 63, 70, 71, 76, 77, 80, 82, 86, 92, 95, 98, 99, 100, 101, 103, 104] [3, 11, 14, 16, 29, 41, 42, 43, 57, 58, 63, 64, 71, 72, 77, 78, 83, 87, 90, 93, 96, 100, 101, 102, 104, 105] [4, 12, 15, 17, 30, 42, 43, 44, 58, 59, 63, 64, 65, 73, 78, 79, 84, 88, 91, 94, 97, 101, 102, 103, 105, 106] [5, 9, 13, 16, 31, 36, 43, 44, 59, 60, 64, 65, 66, 74, 79, 80, 85, 89, 92, 95, 98, 102, 103, 104, 106, 107] [15, 19, 23, 26, 41, 45, 46, 53, 54, 58, 59, 60, 62, 69, 70, 74, 75, 76, 81, 84, 89, 90, 95, 99, 102, 105] [16, 18, 20, 24, 42, 45, 46, 47, 54, 55, 59, 60, 61, 70, 71, 75, 76, 77, 81, 82, 85, 91, 96, 100, 103, 106] [17, 19, 21, 25, 43, 46, 47, 48, 55, 56, 60, 61, 62, 63, 71, 76, 77, 78, 82, 83, 86, 92, 97, 101, 104, 107] [9, 20, 22, 26, 44, 47, 48, 49, 54, 56, 57, 61, 62, 63, 64, 77, 78, 79, 83, 84, 87, 93, 98, 99, 102, 105] [10, 18, 21, 23, 36, 48, 49, 50, 54, 55, 57, 58, 62, 64, 65, 78, 79, 80, 84, 85, 88, 90, 94, 100, 103, 106] [11, 19, 22, 24, 37, 49, 50, 51, 54, 55, 56, 58, 59, 65, 66, 72, 79, 80, 85, 86, 89, 91, 95, 101, 104, 107] [12, 20, 23, 25, 38, 50, 51, 52, 55, 56, 57, 59, 60, 66, 67, 72, 73, 80, 81, 86, 87, 92, 96, 99, 102, 105] [13, 21, 24, 26, 39, 51, 52, 53, 56, 57, 58, 60, 61, 67, 68, 72, 73, 74, 82, 87, 88, 93, 97, 100, 103, 106] [14, 18, 22, 25, 40, 45, 52, 53, 57, 58, 59, 61, 62, 68, 69, 73, 74, 75, 83, 88, 89, 94, 98, 101, 104, 107]

Code ID 108-8-12 · download JSON · raw on GitHub