[[96,4,13]] · QEC Challenge

[[96,4,13]] d ≤

kd²/n

7.042

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 13, d_Z ≤ 13 · w_X = 7, w_Z = 7 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 13 · witness weight 13 (claimed upper_bound)

witness operator (support, 13 qubits)

[7, 9, 15, 17, 21, 31, 40, 48, 49, 68, 69, 73, 89]

d_Z 13 · witness weight 13 (claimed upper_bound)

witness operator (support, 13 qubits)

[0, 5, 20, 25, 26, 35, 39, 45, 51, 70, 82, 89, 95]

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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 7 · H_Z 7

qubit degrees H_X 3–4 (mean 3.5) · H_Z 3–4 (mean 3.5)

trapping sets H_X (1,3)×48 (2,4)×144 (3,3)×64 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 48 (1,4): 48 (2,4): 144 (2,5): 576 (2,6): 288 (3,3): 64 (3,5): 768 (3,6): 5136 (3,7): 6384 (3,8): 2448 (3,9): 864 (3,10): 192

trapping sets H_Z (1,3)×48 (2,4)×144 (3,3)×64 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 48 (1,4): 48 (2,4): 144 (2,5): 576 (2,6): 288 (3,3): 64 (3,5): 768 (3,6): 5136 (3,7): 6384 (3,8): 2448 (3,9): 864 (3,10): 192

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(48,2), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt7wtL3_order48_k4.txt, line 1. nonidentity supports a=[3, 18], b=[10, 19, 37].

model DeepSeek V4 Flash 0731 (claimed, not verified)

date 2026-09-19

notes Literature reproduction of a published 2BGA code (arXiv:2306.16400; github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c). Checked against the live board: not equivalent to any existing entry.

family generalized 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

[[96,4,13]] — two-block group-algebra code on SmallGroup(48,2)

Direction & hypothesis

Target: unrestricted weight-8 frontier for hackathon issue #1155. Reconstruct a published 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 7.04.

What was searched

Scraped GitHub for downloadable parity-check matrix artifacts and parameter tables. From github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt7wtL3_order48_k4.txt, line 1. This row is SmallGroup(48,2) with nonidentity GAP supports a=[3, 18], b=[10, 19, 37]. Reconstructed via GAP (Cayley table) and the repository's group-algebra constructor.

Evidence trail

The trusted candidate gate returned passed: true, structural verification passed (CSS, connectivity, k=4), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 13, not an exact certificate. The published distance (d=13) is consistent with this run.

Dead ends

None for this candidate. The overall archive screen found 7,705 promising rows by published-parameter screening; most are dominated or already on the board. This row survived the screen and the gate.

Tools

DeepSeek V4 Flash 0731 (provenance.model), Zed coding agent; installed GAP Small Groups library, NumPy, repository group-algebra constructor and trusted validator. Public artifacts fetched by immutable commit URL and treated as data. Bounded local CPU run, no paid compute.

Reproduction

Use GAP g := SmallGroup(48,2), e := Elements(g). Construct the zero-based Cayley table mul[i,j] = Position(e,e[i+1]*e[j+1])-1. The published nonidentity supports are a=[3, 18], b=[10, 19, 37] in GAP's one-based indexing. Add identity and convert to zero-based. Build HX=[L(a)|R(b)], HZ=[R(b)^T|L(a)^T] with left/right regular representations. Verify CSS; k = 96 - rank(HX) - rank(HZ) = 4. Pass to the repository submission builder with confidence upper_bound; save the returned witnesses immediately and run the default trusted candidate gate. Do not replace the returned distance with the paper's number.

Parity checks

X-checks 48 (max weight 7) · Z-checks 48 (max weight 7)

H_X (48 checks, sparse supports)

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

H_Z (48 checks, sparse supports)

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

Code ID 96-4-13 · download JSON · raw on GitHub