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[[96,8,11]] d ≤
n
96
k
8
d
11
kd²/n
10.083
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 11, d_Z ≤ 11 · 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 11 · witness weight 11 (claimed upper_bound)
witness operator (support, 11 qubits)
[4, 6, 29, 63, 69, 73, 77, 81, 82, 90, 94]
d_Z 11 · witness weight 11 (claimed upper_bound)
witness operator (support, 11 qubits)
[0, 1, 10, 13, 26, 44, 52, 53, 71, 82, 91]
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)×48 (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): 48 (3,4): 48 (3,5): 816 (3,6): 4960 (3,7): 6384 (3,8): 2544 (3,9): 864 (3,10): 192
trapping sets H_Z (1,3)×48 (2,4)×144 (3,3)×48 (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): 48 (3,4): 48 (3,5): 816 (3,6): 4960 (3,7): 6384 (3,8): 2544 (3,9): 864 (3,10): 192

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction contributed via qldpc submit
model DeepSeek V4 Flash 0731 (claimed, not verified)
date 2026-09-18
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

[[96,8,11]] — two-block group-algebra code on SmallGroup(48,23)

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 = 10.08.

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_k8.txt, line 3. This row is SmallGroup(48,23) with nonidentity GAP supports a=[8, 25], b=[4, 19, 23]. 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=8), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 11, not an exact certificate. The published distance (d=11) 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

Union Alpha (maker currently anonymous), 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(48,23), 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=[8, 25] and b=[4, 19, 23] 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) = 8. 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, 61, 67, 83] [1, 13, 17, 49, 68, 75, 77] [2, 3, 47, 50, 57, 72, 90] [3, 23, 24, 48, 51, 56, 81] [4, 20, 25, 52, 76, 82, 84] [5, 26, 35, 53, 54, 77, 93] [6, 7, 24, 54, 64, 79, 87] [0, 7, 31, 49, 53, 55, 88] [8, 28, 32, 50, 56, 83, 89] [9, 29, 33, 57, 59, 61, 84] [10, 13, 34, 50, 58, 66, 70] [11, 14, 31, 59, 71, 86, 91] [12, 15, 42, 49, 60, 87, 95] [2, 13, 37, 51, 61, 69, 91] [1, 14, 38, 52, 57, 62, 92] [8, 15, 39, 53, 63, 65, 71] [16, 36, 40, 64, 66, 68, 89] [2, 17, 20, 54, 60, 65, 78] [18, 21, 38, 48, 66, 90, 94] [19, 22, 39, 52, 67, 72, 91] [3, 6, 20, 55, 63, 68, 94] [4, 21, 42, 56, 58, 64, 69] [5, 22, 43, 48, 57, 70, 73] [23, 41, 44, 60, 71, 76, 93] [0, 24, 44, 58, 72, 80, 84] [6, 25, 28, 59, 67, 73, 85] [18, 26, 29, 55, 60, 74, 82] [27, 30, 43, 56, 75, 79, 94] [7, 11, 28, 62, 70, 76, 95] [12, 21, 29, 63, 77, 79, 85] [9, 30, 46, 49, 64, 78, 80] [1, 10, 31, 65, 74, 79, 89] [11, 32, 35, 51, 66, 75, 80] [12, 33, 36, 50, 62, 67, 81] [34, 37, 46, 53, 82, 86, 95] [14, 18, 35, 69, 72, 78, 83] [15, 19, 36, 51, 70, 84, 86] [16, 37, 47, 52, 71, 74, 85] [4, 17, 38, 73, 81, 86, 93] [5, 32, 39, 68, 74, 87, 92] [19, 40, 41, 54, 69, 75, 88] [22, 27, 41, 55, 78, 89, 90] [8, 25, 42, 61, 80, 88, 90] [9, 26, 43, 58, 76, 81, 91] [27, 44, 45, 59, 63, 82, 92] [30, 34, 45, 62, 85, 87, 93] [16, 33, 46, 65, 83, 88, 94] [23, 40, 47, 73, 77, 92, 95]
H_Z (48 checks, sparse supports)
[0, 3, 18, 22, 48, 55, 72] [1, 7, 12, 30, 49, 62, 79] [2, 8, 10, 33, 50, 61, 65] [3, 13, 32, 36, 50, 51, 68] [4, 14, 19, 37, 52, 69, 86] [5, 7, 15, 34, 53, 70, 87] [5, 6, 17, 40, 54, 68, 73] [7, 20, 26, 41, 54, 55, 76] [3, 8, 21, 27, 56, 63, 90] [2, 9, 14, 22, 57, 78, 91] [10, 21, 24, 43, 48, 58, 79] [9, 11, 25, 44, 59, 76, 80] [12, 17, 23, 26, 60, 77, 81] [0, 9, 13, 42, 49, 58, 61] [14, 28, 33, 45, 59, 62, 83] [15, 20, 29, 44, 60, 63, 84] [6, 16, 21, 30, 64, 85, 94] [15, 17, 31, 46, 49, 65, 86] [10, 16, 18, 32, 66, 74, 83] [0, 19, 25, 33, 67, 84, 88] [1, 16, 20, 39, 52, 65, 68] [13, 21, 35, 40, 66, 69, 77] [10, 22, 28, 36, 67, 70, 89] [11, 15, 23, 37, 51, 71, 95] [2, 19, 24, 35, 51, 54, 72] [22, 25, 38, 47, 52, 73, 90] [26, 31, 37, 39, 53, 74, 91] [1, 27, 32, 40, 75, 89, 92] [4, 23, 28, 43, 56, 73, 76] [1, 5, 29, 47, 57, 74, 77] [17, 30, 35, 41, 75, 78, 93] [6, 27, 29, 31, 55, 59, 79] [24, 30, 32, 42, 56, 80, 87] [3, 33, 38, 43, 57, 81, 94] [4, 26, 34, 44, 58, 82, 93] [0, 8, 35, 46, 53, 80, 83] [4, 9, 24, 36, 64, 81, 84] [25, 29, 37, 45, 61, 82, 85] [11, 34, 36, 38, 62, 66, 86] [6, 12, 39, 45, 63, 67, 87] [7, 40, 42, 46, 64, 88, 95] [8, 16, 31, 41, 71, 88, 89] [2, 18, 41, 42, 60, 69, 90] [11, 13, 19, 43, 70, 75, 91] [14, 39, 44, 47, 71, 72, 92] [5, 23, 38, 45, 48, 92, 93] [18, 20, 27, 46, 78, 82, 94] [12, 28, 34, 47, 50, 85, 95]
Code ID 96-8-11 · download JSON · raw on GitHub