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[[128,2,16]] d ≤
n
128
k
2
d
16
kd²/n
4.0
w
6
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 16, d_Z ≤ 16 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[8, 16, 19, 42, 52, 58, 59, 62, 70, 76, 78, 93, 98, 101, 102, 112]
d_Z 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[3, 14, 23, 24, 31, 32, 55, 60, 70, 87, 89, 93, 98, 104, 110, 119]
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 6 · H_Z 6
qubit degrees H_X 2–4 (mean 3.0) · H_Z 2–4 (mean 3.0)
trapping sets H_X (1,2)×64 (2,2)×64 (3,2)×64 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 64 (1,4): 64 (2,2): 64 (2,4): 512 (2,6): 384 (3,2): 64 (3,4): 2112 (3,6): 5952 (3,8): 3456 (3,10): 256
trapping sets H_Z (1,2)×64 (2,2)×64 (3,2)×64 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 64 (1,4): 64 (2,2): 64 (2,4): 512 (2,6): 384 (3,2): 64 (3,4): 2112 (3,6): 5952 (3,8): 3456 (3,10): 256

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 ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[128,2,16]] — two-block group-algebra code on SmallGroup(64,51)

Direction & hypothesis

Target: unrestricted weight-6 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 = 4.00.

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_wt6wtL2_order64_k2.txt, line 149. This row is SmallGroup(64,51) with nonidentity GAP supports a=[2], b=[8, 11, 40]. 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=2), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 16, not an exact certificate. The published distance (d=16) 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(64,51), 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=[2] and b=[8, 11, 40] 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 = 128 - rank(HX) - rank(HZ) = 2. 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 64 (max weight 6) · Z-checks 64 (max weight 6)
H_X (64 checks, sparse supports)
[0, 61, 64, 81, 113, 121] [0, 1, 65, 79, 85, 91] [2, 63, 66, 97, 106, 125] [1, 3, 67, 83, 89, 95] [4, 8, 68, 102, 108, 114] [5, 26, 69, 82, 122, 125] [6, 48, 70, 90, 104, 127] [2, 7, 64, 71, 78, 107] [3, 8, 72, 93, 98, 104] [4, 9, 73, 100, 105, 112] [5, 10, 64, 74, 92, 101] [6, 11, 66, 69, 75, 114] [7, 12, 65, 76, 88, 99] [13, 22, 72, 77, 107, 116] [14, 42, 78, 90, 98, 121] [15, 57, 79, 112, 118, 122] [9, 16, 70, 80, 110, 115] [10, 17, 65, 81, 84, 111] [11, 18, 71, 74, 82, 105] [19, 27, 72, 83, 103, 123] [20, 28, 84, 86, 91, 120] [21, 49, 67, 85, 106, 112] [12, 22, 67, 86, 97, 109] [13, 23, 68, 87, 99, 121] [14, 24, 69, 79, 88, 108] [15, 25, 70, 89, 101, 123] [16, 26, 75, 90, 117, 120] [17, 27, 67, 91, 94, 118] [18, 28, 76, 81, 92, 115] [19, 29, 68, 93, 113, 119] [20, 30, 77, 83, 94, 125] [21, 31, 70, 72, 78, 95] [23, 32, 73, 79, 96, 109] [24, 33, 74, 89, 97, 100] [25, 34, 75, 98, 111, 119] [35, 43, 91, 99, 108, 117] [36, 44, 92, 100, 123, 126] [37, 58, 76, 101, 113, 127] [29, 38, 73, 85, 102, 124] [30, 39, 64, 87, 93, 103] [31, 40, 68, 75, 88, 104] [41, 50, 80, 92, 105, 107] [32, 42, 80, 89, 106, 116] [33, 43, 81, 98, 107, 110] [34, 44, 82, 108, 118, 124] [35, 45, 83, 100, 109, 122] [36, 46, 84, 110, 119, 127] [37, 47, 66, 85, 86, 111] [38, 48, 80, 95, 112, 126] [39, 49, 65, 96, 102, 113] [40, 50, 73, 82, 97, 114] [41, 51, 84, 90, 99, 115] [45, 52, 93, 101, 110, 116] [46, 53, 66, 94, 117, 124] [47, 54, 71, 77, 95, 118] [55, 59, 86, 96, 114, 119] [51, 56, 69, 94, 109, 120] [52, 57, 102, 111, 117, 121] [53, 58, 71, 103, 122, 126] [54, 59, 76, 87, 104, 123] [55, 60, 77, 105, 106, 124] [56, 61, 74, 103, 116, 125] [60, 62, 78, 87, 115, 126] [62, 63, 88, 96, 120, 127]
H_Z (64 checks, sparse supports)
[0, 7, 10, 39, 64, 65] [1, 12, 17, 49, 65, 67] [2, 11, 47, 53, 66, 71] [3, 21, 22, 27, 67, 72] [4, 23, 29, 40, 68, 73] [5, 11, 24, 56, 69, 74] [6, 16, 25, 31, 70, 75] [7, 18, 54, 58, 71, 76] [8, 13, 19, 31, 68, 72] [9, 32, 38, 50, 73, 80] [10, 18, 33, 61, 74, 81] [11, 26, 34, 40, 75, 82] [12, 28, 37, 59, 76, 86] [13, 30, 54, 60, 77, 87] [7, 14, 31, 62, 78, 88] [1, 15, 24, 32, 79, 89] [16, 41, 42, 48, 80, 90] [0, 17, 28, 43, 81, 91] [5, 18, 44, 50, 82, 92] [3, 19, 30, 45, 83, 93] [17, 20, 46, 51, 84, 94] [1, 21, 38, 47, 85, 95] [20, 22, 47, 55, 77, 86] [23, 39, 59, 62, 87, 96] [12, 24, 40, 63, 88, 97] [3, 25, 33, 42, 89, 98] [6, 14, 26, 51, 69, 90] [1, 20, 27, 35, 83, 91] [10, 28, 36, 41, 84, 92] [8, 29, 39, 52, 93, 102] [27, 30, 53, 56, 94, 103] [3, 31, 48, 54, 95, 104] [32, 49, 55, 63, 96, 106] [2, 22, 33, 50, 97, 107] [8, 14, 34, 43, 98, 108] [12, 23, 35, 51, 99, 109] [9, 33, 36, 45, 100, 110] [10, 25, 37, 52, 101, 111] [4, 38, 49, 57, 102, 112] [19, 39, 58, 61, 103, 113] [6, 8, 40, 59, 104, 114] [9, 18, 41, 60, 105, 115] [2, 21, 42, 60, 78, 106] [7, 13, 41, 43, 99, 107] [4, 24, 35, 44, 100, 108] [22, 32, 45, 56, 109, 116] [16, 43, 46, 52, 110, 117] [17, 34, 47, 57, 111, 118] [9, 15, 21, 48, 70, 112] [0, 29, 37, 49, 85, 113] [4, 11, 50, 55, 105, 114] [16, 28, 51, 62, 115, 120] [13, 42, 52, 61, 116, 121] [26, 35, 53, 57, 117, 122] [15, 27, 44, 54, 118, 123] [29, 34, 46, 55, 119, 124] [20, 26, 56, 63, 120, 125] [0, 14, 23, 57, 79, 121] [5, 15, 45, 58, 101, 122] [19, 25, 36, 59, 119, 123] [38, 44, 53, 60, 124, 126] [2, 5, 30, 61, 64, 125] [36, 48, 58, 62, 126, 127] [6, 37, 46, 63, 66, 127]
Code ID 128-2-16 · download JSON · raw on GitHub