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[[128,40,2]] d =
n
128
k
40
d
2
kd²/n
1.25
w
6
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · 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 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[4, 35]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[16, 32]
certificate exact, d = 2 · scipy/HiGHS MILP
X: no logical < 2 exists; Z: no logical < 2 exists

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 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 (2,0)×32 (3,2)×512 (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 (2,0): 32 (2,2): 64 (2,4): 576 (3,2): 512 (3,4): 2720 (3,6): 1376 (3,8): 128
trapping sets H_Z (1,2)×64 (2,0)×64 (3,0)×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 (2,0): 64 (2,2): 64 (2,4): 384 (2,6): 256 (3,0): 64 (3,2): 224 (3,4): 1888 (3,6): 3200 (3,8): 1792

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on SmallGroup(64,4), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt6wtL2_order64_k40.txt, line 1. nonidentity supports a=[3], b=[2, 3, 8].
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 ≤ 6 (computed)

How this code was found

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

[[128,40,2]] — two-block group-algebra code on SmallGroup(64,4)

Direction & hypothesis

Target: unrestricted weight-6 frontier for hackathon issue #1155. Reconstruct a published weight-6 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 1.25.

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_k40.txt, line 1. This row is SmallGroup(64,4) with nonidentity GAP supports a=[3], b=[2, 3, 8]. 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=40), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 2, not an exact certificate. The published distance (d=2) 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(64,4), 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], b=[2, 3, 8] 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) = 40. 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, 2, 64, 66, 94, 122] [1, 22, 64, 65, 71, 97] [0, 2, 64, 66, 94, 122] [3, 12, 67, 76, 124, 125] [4, 13, 65, 68, 99, 107] [5, 14, 69, 78, 115, 127] [6, 15, 70, 73, 79, 106] [7, 8, 64, 65, 71, 97] [7, 8, 66, 72, 81, 86] [9, 42, 68, 73, 96, 109] [10, 43, 69, 74, 76, 88] [11, 44, 70, 75, 89, 118] [3, 12, 67, 76, 124, 125] [4, 13, 74, 77, 83, 86] [5, 14, 69, 78, 115, 127] [6, 15, 70, 73, 79, 106] [16, 32, 71, 80, 91, 116] [17, 33, 81, 97, 110, 113] [18, 34, 82, 98, 109, 112] [19, 35, 74, 77, 83, 86] [20, 36, 75, 84, 119, 123] [21, 37, 85, 93, 101, 121] [1, 22, 66, 72, 81, 86] [23, 26, 83, 87, 93, 116] [24, 27, 69, 74, 76, 88] [25, 28, 70, 75, 89, 118] [23, 26, 90, 99, 102, 121] [24, 27, 67, 78, 91, 107] [25, 28, 79, 92, 104, 108] [29, 57, 83, 87, 93, 116] [30, 58, 84, 94, 117, 124] [31, 59, 85, 95, 98, 111] [16, 32, 72, 88, 96, 102] [17, 33, 81, 97, 110, 113] [18, 34, 82, 98, 109, 112] [19, 35, 65, 68, 99, 107] [20, 36, 95, 100, 105, 108] [21, 37, 85, 93, 101, 121] [38, 52, 72, 88, 96, 102] [39, 53, 89, 103, 114, 126] [40, 54, 87, 90, 104, 118] [41, 55, 95, 100, 105, 108] [9, 42, 77, 80, 106, 112] [10, 43, 67, 78, 91, 107] [11, 44, 79, 92, 104, 108] [45, 48, 68, 73, 96, 109] [46, 49, 105, 110, 115, 126] [47, 50, 85, 95, 98, 111] [45, 48, 77, 80, 106, 112] [46, 49, 113, 119, 120, 127] [47, 50, 82, 101, 114, 123] [51, 63, 105, 110, 115, 126] [38, 52, 71, 80, 91, 116] [39, 53, 92, 111, 117, 120] [40, 54, 87, 90, 104, 118] [41, 55, 75, 84, 119, 123] [56, 62, 92, 111, 117, 120] [29, 57, 90, 99, 102, 121] [30, 58, 100, 103, 122, 125] [31, 59, 82, 101, 114, 123] [60, 61, 84, 94, 117, 124] [60, 61, 100, 103, 122, 125] [56, 62, 89, 103, 114, 126] [51, 63, 113, 119, 120, 127]
H_Z (64 checks, sparse supports)
[0, 1, 2, 7, 64, 66] [1, 4, 7, 35, 65, 86] [0, 2, 8, 22, 64, 66] [3, 12, 27, 43, 67, 76] [4, 9, 35, 45, 68, 77] [5, 10, 14, 24, 69, 78] [6, 11, 15, 25, 70, 79] [1, 7, 16, 52, 71, 72] [8, 22, 32, 38, 71, 72] [6, 9, 15, 45, 73, 106] [10, 13, 19, 24, 74, 107] [11, 20, 25, 55, 75, 108] [3, 10, 12, 24, 67, 76] [13, 19, 42, 48, 68, 77] [5, 14, 27, 43, 69, 78] [6, 15, 28, 44, 70, 79] [16, 42, 48, 52, 80, 96] [8, 17, 22, 33, 81, 97] [18, 34, 50, 59, 82, 98] [13, 19, 23, 29, 83, 99] [20, 30, 55, 60, 84, 100] [21, 31, 37, 47, 85, 101] [8, 13, 19, 22, 65, 86] [23, 29, 40, 54, 87, 90] [10, 24, 32, 38, 88, 91] [11, 25, 39, 62, 89, 92] [26, 40, 54, 57, 87, 90] [16, 27, 43, 52, 88, 91] [28, 44, 53, 56, 89, 92] [21, 23, 29, 37, 93, 121] [0, 2, 30, 60, 94, 122] [31, 36, 41, 47, 95, 123] [9, 32, 38, 45, 80, 96] [1, 7, 17, 33, 81, 97] [18, 31, 34, 47, 82, 98] [4, 26, 35, 57, 83, 99] [36, 41, 58, 61, 84, 100] [21, 37, 50, 59, 85, 101] [26, 32, 38, 57, 102, 116] [39, 58, 61, 62, 103, 117] [28, 40, 44, 54, 104, 118] [36, 41, 46, 51, 105, 119] [6, 15, 42, 48, 73, 106] [4, 27, 35, 43, 74, 107] [28, 36, 41, 44, 75, 108] [9, 18, 34, 45, 109, 112] [17, 33, 46, 51, 110, 113] [31, 47, 53, 56, 111, 114] [18, 34, 42, 48, 109, 112] [17, 33, 49, 63, 110, 113] [39, 50, 59, 62, 111, 114] [5, 14, 46, 51, 115, 127] [16, 23, 29, 52, 102, 116] [30, 53, 56, 60, 103, 117] [11, 25, 40, 54, 104, 118] [20, 49, 55, 63, 105, 119] [49, 53, 56, 63, 120, 126] [21, 26, 37, 57, 93, 121] [0, 2, 58, 61, 94, 122] [20, 50, 55, 59, 95, 123] [3, 12, 30, 60, 124, 125] [3, 12, 58, 61, 124, 125] [39, 46, 51, 62, 120, 126] [5, 14, 49, 63, 115, 127]
Code ID 128-40-2 · download JSON · raw on GitHub