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[[108,6,6]] d =
n
108
k
6
d
6
kd²/n
2.0
w
5
X/Z
1
g
0.0062
r
4.2426
layers
2
swaps
262

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Distance

X/Z asymmetry 1 · d_X = 6, d_Z = 6 · w_X = 5, w_Z = 5 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[7, 10, 34, 42, 73, 96]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[0, 53, 54, 58, 85, 107]
certificate exact, d = 6 · scipy/HiGHS MILP
X: no logical < 6 exists; Z: no logical < 6 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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 5 · H_Z 5
qubit degrees H_X 2–3 (mean 2.5) · H_Z 2–3 (mean 2.5)
trapping sets H_X (1,2)×54 (2,2)×54 (3,2)×126 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 54 (1,3): 54 (2,2): 54 (2,3): 324 (2,4): 162 (3,2): 126 (3,3): 972 (3,4): 1566 (3,5): 810 (3,6): 324 (3,7): 54
trapping sets H_Z (1,2)×54 (2,2)×54 (3,2)×126 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 54 (1,3): 54 (2,2): 54 (2,3): 324 (2,4): 162 (3,2): 126 (3,3): 972 (3,4): 1566 (3,5): 810 (3,6): 324 (3,7): 54
witness diameter X 3.1623 · Z 2.0 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
r = 4.243
X checkZ checkqubit site (54)2 qubits stacked (2 layers)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release
routing cost 262 nearest-neighbor SWAPs per round in total, at most 7 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on SmallGroup(54,4), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt5wtL2_order54_k6.txt, line 3. nonidentity supports a=[7], b=[6, 22].
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 2D-local bilayer (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

[[108,6,6]] — two-block group-algebra code on SmallGroup(54,4)

Direction & hypothesis

Target: unrestricted weight-6 frontier for hackathon issue #1155. Reconstruct a published weight-5 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 2.00.

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_wt5wtL2_order54_k6.txt, line 3. This row is SmallGroup(54,4) with nonidentity GAP supports a=[7], b=[6, 22]. 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=6), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 6, not an exact certificate. The published distance (d=6) 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(54,4), 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=[7], b=[6, 22] 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 = 108 - rank(HX) - rank(HZ) = 6. 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 54 (max weight 5) · Z-checks 54 (max weight 5)
H_X (54 checks, sparse supports)
[0, 17, 54, 94, 102] [1, 11, 55, 88, 105] [2, 29, 55, 56, 106] [1, 3, 57, 68, 78] [4, 32, 58, 76, 107] [5, 22, 54, 59, 107] [0, 6, 60, 62, 85] [7, 25, 61, 83, 106] [8, 40, 59, 62, 67] [5, 9, 60, 63, 90] [10, 43, 64, 73, 88] [6, 11, 65, 81, 92] [7, 12, 56, 66, 96] [13, 45, 67, 91, 103] [14, 34, 56, 68, 73] [2, 15, 57, 69, 96] [16, 37, 67, 70, 94] [3, 17, 71, 74, 98] [4, 18, 59, 72, 90] [19, 39, 73, 97, 100] [14, 20, 69, 74, 80] [21, 49, 54, 75, 85] [15, 22, 71, 76, 101] [16, 23, 62, 77, 87] [24, 51, 61, 78, 100] [18, 25, 63, 79, 104] [19, 26, 64, 80, 82] [8, 27, 63, 81, 87] [28, 46, 55, 78, 82] [9, 29, 65, 83, 104] [10, 30, 68, 80, 84] [31, 48, 58, 85, 103] [12, 32, 69, 86, 101] [13, 33, 70, 75, 87] [27, 34, 83, 88, 93] [28, 35, 57, 89, 98] [36, 53, 58, 70, 90] [30, 37, 74, 91, 99] [31, 38, 72, 75, 92] [33, 39, 77, 93, 95] [20, 40, 76, 94, 99] [21, 41, 60, 92, 95] [42, 52, 61, 64, 96] [23, 43, 81, 93, 97] [24, 44, 66, 82, 98] [26, 45, 84, 89, 99] [41, 46, 65, 100, 105] [42, 47, 66, 84, 101] [44, 48, 86, 89, 102] [35, 49, 71, 102, 103] [36, 50, 72, 77, 104] [38, 51, 79, 95, 105] [50, 52, 79, 97, 106] [47, 53, 86, 91, 107]
H_Z (54 checks, sparse supports)
[0, 5, 21, 54, 60] [1, 2, 28, 55, 57] [2, 12, 14, 56, 69] [3, 15, 35, 57, 71] [4, 31, 36, 58, 72] [5, 8, 18, 59, 63] [6, 9, 41, 60, 65] [7, 24, 42, 61, 66] [6, 8, 23, 62, 81] [9, 25, 27, 63, 83] [10, 26, 42, 64, 84] [11, 29, 46, 55, 65] [12, 44, 47, 66, 86] [8, 13, 16, 67, 87] [3, 14, 30, 68, 74] [15, 20, 32, 69, 76] [16, 33, 36, 70, 77] [17, 22, 49, 54, 71] [18, 38, 50, 72, 79] [10, 14, 19, 73, 80] [17, 20, 37, 74, 94] [21, 33, 38, 75, 95] [4, 22, 40, 59, 76] [23, 39, 50, 77, 97] [3, 24, 28, 78, 98] [25, 51, 52, 61, 79] [20, 26, 30, 80, 99] [11, 27, 43, 81, 88] [26, 28, 44, 82, 89] [7, 29, 34, 56, 83] [30, 45, 47, 84, 91] [6, 21, 31, 85, 92] [32, 48, 53, 58, 86] [23, 27, 33, 87, 93] [1, 10, 34, 68, 88] [35, 45, 48, 89, 103] [9, 18, 36, 90, 104] [13, 37, 53, 70, 91] [11, 38, 41, 92, 105] [34, 39, 43, 73, 93] [0, 16, 40, 62, 94] [39, 41, 51, 95, 100] [12, 15, 42, 96, 101] [19, 43, 52, 64, 97] [17, 35, 44, 98, 102] [37, 40, 45, 67, 99] [19, 24, 46, 82, 100] [22, 32, 47, 101, 107] [0, 48, 49, 85, 102] [13, 31, 49, 75, 103] [25, 29, 50, 104, 106] [1, 46, 51, 78, 105] [2, 7, 52, 96, 106] [4, 5, 53, 90, 107]
Code ID 108-6-6 · download JSON · raw on GitHub