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[[126,20,10]] d ≤
n
126
k
20
d
10
kd²/n
15.873
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 10, d_Z ≤ 10 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[12, 29, 39, 40, 49, 59, 60, 70, 75, 122]
d_Z 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[14, 43, 59, 73, 77, 101, 103, 112, 113, 122]
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 8 · H_Z 8
qubit degrees H_X 3–5 (mean 4.0) · H_Z 3–5 (mean 4.0)
trapping sets H_X (1,3)×63 (2,4)×189 (3,3)×63 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 63 (1,5): 63 (2,4): 189 (2,6): 945 (2,8): 630 (3,3): 63 (3,5): 819 (3,7): 9639 (3,9): 19236 (3,11): 8694 (3,13): 630
trapping sets H_Z (1,3)×63 (2,4)×189 (3,3)×63 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 63 (1,5): 63 (2,4): 189 (2,6): 945 (2,8): 630 (3,3): 63 (3,5): 819 (3,7): 9639 (3,9): 19236 (3,11): 8694 (3,13): 630

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

[[126,20,10]] — two-block group-algebra code on SmallGroup(63,3)

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 (upstream main 72d7db93). Efficiency kd²/n = 20·100/126 = 15.87.

What was searched

Scraped GitHub for downloadable parity-check matrix artifacts and parameter tables. From github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL3_order63_k20.txt, line 6. This row is SmallGroup(63,3) with nonidentity GAP supports a=[9,34], b=[2,4,25,47]. 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=20), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 10, not an exact certificate. The published distance (d=10) 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(63,3), 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=[9,34] and b=[2,4,25,47] 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 = 126 - rank(HX) - rank(HZ) = 20. 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 63 (max weight 8) · Z-checks 63 (max weight 8)
H_X (63 checks, sparse supports)
[0, 34, 58, 63, 67, 106, 107, 124] [1, 13, 56, 63, 64, 112, 113, 125] [2, 42, 44, 65, 73, 113, 114, 115] [3, 7, 43, 63, 66, 84, 101, 115] [4, 28, 45, 64, 67, 117, 118, 121] [5, 21, 41, 65, 68, 118, 119, 120] [6, 22, 61, 64, 69, 89, 90, 120] [7, 26, 52, 70, 81, 98, 120, 121] [0, 8, 51, 65, 69, 71, 109, 121] [9, 15, 52, 65, 66, 72, 74, 102] [10, 29, 36, 68, 73, 107, 122, 123] [11, 37, 53, 67, 74, 78, 95, 123] [6, 12, 49, 70, 75, 104, 123, 124] [13, 30, 50, 68, 74, 76, 97, 124] [12, 14, 31, 66, 68, 69, 77, 108] [2, 15, 35, 70, 76, 78, 107, 116] [3, 16, 58, 70, 71, 79, 82, 86] [2, 17, 24, 71, 72, 80, 110, 119] [18, 20, 37, 75, 81, 110, 115, 125] [19, 38, 45, 66, 73, 82, 103, 125] [20, 46, 59, 69, 73, 74, 83, 96] [14, 21, 57, 75, 82, 84, 105, 113] [1, 22, 39, 71, 75, 76, 85, 92] [21, 23, 40, 76, 77, 86, 98, 122] [8, 24, 44, 63, 78, 87, 90, 94] [7, 9, 25, 78, 79, 88, 104, 117] [8, 26, 33, 75, 79, 80, 83, 89] [27, 29, 46, 71, 81, 90, 111, 118] [28, 47, 53, 76, 80, 81, 82, 91] [29, 54, 62, 82, 83, 92, 104, 114] [5, 23, 30, 64, 78, 84, 93, 100] [6, 31, 48, 84, 85, 94, 106, 110] [30, 32, 49, 72, 81, 85, 86, 95] [0, 16, 33, 66, 87, 96, 112, 122] [15, 17, 34, 64, 87, 88, 91, 97] [16, 35, 42, 88, 89, 93, 98, 118] [36, 38, 54, 67, 84, 88, 90, 99] [37, 55, 59, 90, 91, 98, 100, 112] [18, 38, 60, 70, 77, 91, 92, 101] [13, 32, 39, 69, 93, 102, 114, 117] [14, 40, 56, 67, 79, 93, 94, 103] [39, 41, 57, 94, 95, 99, 104, 107] [3, 25, 42, 68, 72, 96, 99, 105] [24, 26, 43, 77, 96, 97, 106, 123] [25, 44, 51, 92, 97, 98, 107, 111] [45, 47, 60, 74, 99, 106, 108, 119] [4, 46, 62, 63, 85, 99, 100, 109] [10, 27, 47, 87, 100, 101, 110, 113] [22, 41, 48, 73, 77, 87, 102, 111] [23, 49, 61, 83, 102, 103, 112, 115] [5, 48, 50, 80, 103, 104, 113, 116] [9, 34, 51, 80, 85, 105, 114, 125] [33, 35, 52, 95, 100, 105, 106, 115] [10, 53, 55, 65, 83, 93, 108, 116] [11, 18, 54, 72, 108, 109, 117, 120] [19, 36, 55, 86, 105, 109, 110, 118] [31, 50, 56, 86, 91, 111, 119, 121] [12, 32, 57, 88, 101, 111, 112, 120] [17, 43, 58, 89, 103, 108, 114, 121] [4, 19, 59, 79, 92, 116, 122, 124] [20, 27, 60, 89, 94, 116, 117, 123] [1, 40, 61, 95, 96, 109, 119, 124] [11, 28, 62, 97, 101, 102, 122, 125]
H_Z (63 checks, sparse supports)
[0, 1, 3, 24, 46, 63, 71, 96] [1, 4, 6, 30, 34, 64, 85, 124] [2, 5, 8, 9, 53, 65, 78, 80] [3, 9, 14, 19, 33, 66, 79, 105] [0, 4, 11, 36, 40, 67, 109, 122] [5, 10, 13, 14, 42, 68, 93, 113] [6, 8, 14, 20, 39, 69, 75, 94] [7, 12, 15, 16, 38, 66, 70, 88] [8, 16, 17, 22, 27, 71, 87, 89] [9, 17, 32, 42, 54, 72, 88, 114] [2, 10, 19, 20, 48, 73, 110, 116] [9, 11, 13, 20, 45, 74, 117, 125] [12, 18, 21, 22, 26, 75, 77, 120] [13, 15, 22, 23, 28, 64, 76, 102] [14, 23, 38, 43, 48, 77, 84, 103] [11, 15, 24, 25, 30, 72, 78, 97] [16, 25, 26, 40, 59, 79, 96, 98] [17, 26, 28, 50, 51, 80, 97, 121] [7, 18, 27, 28, 32, 81, 101, 117] [16, 19, 21, 28, 29, 82, 118, 122] [20, 26, 29, 49, 53, 81, 83, 123] [3, 21, 30, 31, 36, 68, 84, 86] [22, 31, 32, 46, 51, 69, 85, 111] [16, 23, 32, 55, 56, 86, 93, 112] [24, 33, 34, 47, 48, 80, 87, 106] [25, 34, 35, 36, 57, 88, 105, 107] [6, 26, 35, 58, 60, 70, 89, 106] [6, 24, 27, 36, 37, 90, 110, 123] [28, 34, 37, 38, 56, 67, 91, 125] [22, 29, 38, 44, 59, 73, 90, 92] [30, 35, 39, 40, 53, 76, 93, 95] [24, 31, 40, 41, 60, 77, 94, 119] [11, 32, 41, 52, 61, 95, 102, 120] [20, 33, 42, 43, 61, 89, 96, 115] [13, 34, 43, 44, 62, 63, 97, 114] [7, 23, 35, 37, 44, 78, 98, 115] [36, 41, 42, 45, 46, 73, 99, 118] [30, 37, 46, 47, 52, 74, 81, 100] [3, 38, 47, 57, 62, 82, 99, 101] [9, 39, 48, 49, 62, 85, 102, 104] [19, 40, 49, 50, 58, 86, 103, 124] [12, 25, 29, 41, 50, 68, 104, 111] [21, 42, 51, 52, 55, 65, 98, 105] [0, 31, 43, 45, 52, 66, 106, 121] [0, 10, 15, 41, 44, 65, 87, 107] [14, 45, 53, 54, 58, 67, 82, 108] [8, 46, 54, 55, 61, 83, 90, 109] [17, 18, 31, 47, 55, 91, 108, 110] [27, 44, 48, 56, 57, 94, 111, 113] [1, 33, 37, 49, 57, 75, 95, 112] [1, 2, 21, 47, 50, 76, 113, 119] [2, 29, 39, 51, 58, 71, 107, 114] [2, 3, 18, 49, 52, 70, 72, 115] [15, 50, 53, 59, 60, 74, 91, 116] [4, 25, 39, 54, 60, 92, 99, 117] [4, 5, 27, 35, 55, 100, 116, 118] [5, 17, 45, 56, 61, 64, 103, 119] [5, 6, 7, 54, 57, 84, 104, 120] [4, 7, 8, 56, 58, 63, 79, 121] [10, 23, 33, 59, 62, 83, 100, 122] [10, 11, 12, 43, 60, 101, 108, 123] [0, 12, 13, 59, 61, 69, 112, 124] [1, 18, 19, 51, 62, 92, 109, 125]
Code ID 126-20-10 · download JSON · raw on GitHub