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[[128,48,2]] d =
n
128
k
48
d
2
kd²/n
1.5
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)
[124, 125]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[17, 33]
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)×1024 (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): 128 (2,4): 448 (3,2): 1024 (3,4): 1536 (3,6): 960 (3,8): 128
trapping sets H_Z (1,2)×64 (2,0)×64 (3,0)×128 (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): 128 (2,4): 256 (2,6): 256 (3,0): 128 (3,2): 192 (3,4): 1728 (3,6): 2304 (3,8): 1536

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on SmallGroup(64,29), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt6wtL2_order64_k48.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

[[128,48,2]] — two-block group-algebra code on SmallGroup(64,29)

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 = 1.50.

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_k48.txt, line 1. This row is SmallGroup(64,29) 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=48), 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(64,29), 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) = 48. 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, 115, 127] [1, 22, 64, 65, 71, 76] [0, 2, 64, 66, 115, 127] [3, 12, 67, 76, 124, 125] [4, 13, 65, 68, 77, 86] [5, 14, 69, 73, 78, 106] [6, 15, 70, 79, 93, 121] [7, 8, 64, 65, 71, 76] [7, 8, 66, 67, 72, 86] [9, 42, 68, 73, 87, 96] [10, 43, 69, 74, 88, 97] [11, 44, 70, 75, 89, 98] [3, 12, 67, 76, 124, 125] [4, 13, 65, 68, 77, 86] [5, 14, 69, 73, 78, 106] [6, 15, 70, 79, 93, 121] [16, 32, 71, 72, 80, 96] [17, 33, 81, 87, 90, 97] [18, 34, 82, 98, 109, 112] [19, 35, 74, 83, 99, 107] [20, 36, 75, 84, 100, 108] [21, 37, 85, 94, 101, 122] [1, 22, 66, 67, 72, 86] [23, 26, 68, 73, 87, 96] [24, 27, 69, 74, 88, 97] [25, 28, 70, 75, 89, 98] [23, 26, 77, 80, 90, 106] [24, 27, 78, 81, 91, 107] [25, 28, 79, 82, 92, 108] [29, 57, 83, 93, 109, 116] [30, 58, 84, 94, 110, 117] [31, 59, 85, 95, 111, 118] [16, 32, 71, 72, 80, 96] [17, 33, 81, 87, 90, 97] [18, 34, 82, 98, 109, 112] [19, 35, 74, 83, 99, 107] [20, 36, 75, 84, 100, 108] [21, 37, 85, 94, 101, 122] [38, 52, 88, 91, 102, 116] [39, 53, 89, 92, 103, 117] [40, 54, 104, 110, 113, 118] [41, 55, 95, 105, 119, 123] [9, 42, 77, 80, 90, 106] [10, 43, 78, 81, 91, 107] [11, 44, 79, 82, 92, 108] [45, 48, 83, 93, 109, 116] [46, 49, 84, 94, 110, 117] [47, 50, 85, 95, 111, 118] [45, 48, 99, 102, 112, 121] [46, 49, 100, 103, 113, 122] [47, 50, 101, 104, 114, 123] [51, 63, 105, 115, 124, 126] [38, 52, 88, 91, 102, 116] [39, 53, 89, 92, 103, 117] [40, 54, 104, 110, 113, 118] [41, 55, 95, 105, 119, 123] [56, 62, 111, 114, 120, 126] [29, 57, 99, 102, 112, 121] [30, 58, 100, 103, 113, 122] [31, 59, 101, 104, 114, 123] [60, 61, 105, 115, 124, 126] [60, 61, 119, 120, 125, 127] [56, 62, 111, 114, 120, 126] [51, 63, 119, 120, 125, 127]
H_Z (64 checks, sparse supports)
[0, 1, 2, 7, 64, 66] [1, 4, 7, 13, 65, 86] [0, 2, 8, 22, 64, 66] [3, 8, 12, 22, 67, 76] [4, 9, 13, 23, 68, 77] [5, 10, 14, 24, 69, 78] [6, 11, 15, 25, 70, 79] [1, 7, 16, 32, 71, 72] [8, 16, 22, 32, 71, 72] [5, 9, 14, 23, 73, 106] [10, 19, 24, 35, 74, 107] [11, 20, 25, 36, 75, 108] [1, 3, 7, 12, 67, 76] [4, 13, 26, 42, 68, 77] [5, 14, 27, 43, 69, 78] [6, 15, 28, 44, 70, 79] [16, 26, 32, 42, 80, 96] [17, 27, 33, 43, 81, 97] [18, 28, 34, 44, 82, 98] [19, 29, 35, 45, 83, 99] [20, 30, 36, 46, 84, 100] [21, 31, 37, 47, 85, 101] [4, 8, 13, 22, 65, 86] [9, 17, 23, 33, 87, 90] [10, 24, 38, 52, 88, 91] [11, 25, 39, 53, 89, 92] [17, 26, 33, 42, 87, 90] [27, 38, 43, 52, 88, 91] [28, 39, 44, 53, 89, 92] [6, 15, 29, 45, 93, 121] [21, 30, 37, 46, 94, 122] [31, 41, 47, 55, 95, 123] [9, 16, 23, 32, 80, 96] [10, 17, 24, 33, 81, 97] [11, 18, 25, 34, 82, 98] [19, 35, 48, 57, 83, 99] [20, 36, 49, 58, 84, 100] [21, 37, 50, 59, 85, 101] [38, 48, 52, 57, 102, 116] [39, 49, 53, 58, 103, 117] [40, 50, 54, 59, 104, 118] [41, 51, 55, 60, 105, 119] [5, 14, 26, 42, 73, 106] [19, 27, 35, 43, 74, 107] [20, 28, 36, 44, 75, 108] [18, 29, 34, 45, 109, 112] [30, 40, 46, 54, 110, 113] [31, 47, 56, 62, 111, 114] [18, 34, 48, 57, 109, 112] [40, 49, 54, 58, 110, 113] [50, 56, 59, 62, 111, 114] [0, 2, 51, 60, 115, 127] [29, 38, 45, 52, 102, 116] [30, 39, 46, 53, 103, 117] [31, 40, 47, 54, 104, 118] [41, 55, 61, 63, 105, 119] [56, 61, 62, 63, 120, 126] [6, 15, 48, 57, 93, 121] [21, 37, 49, 58, 94, 122] [41, 50, 55, 59, 95, 123] [3, 12, 51, 60, 124, 125] [3, 12, 61, 63, 124, 125] [51, 56, 60, 62, 120, 126] [0, 2, 61, 63, 115, 127]
Code ID 128-48-2 · download JSON · raw on GitHub