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[[108,16,2]] d =
n
108
k
16
d
2
kd²/n
0.593
w
5
X/Z
1
g
0.0009
r
5.099
layers
2
swaps
349

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Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · 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 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[91, 97]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[46, 49]
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 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 (2,0)×27 (3,3)×810 (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 (2,0): 27 (2,3): 324 (2,4): 162 (3,3): 810 (3,4): 1782 (3,5): 648 (3,6): 324 (3,7): 54
trapping sets H_Z (1,2)×54 (2,0)×27 (3,3)×810 (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 (2,0): 27 (2,3): 324 (2,4): 162 (3,3): 810 (3,4): 1782 (3,5): 648 (3,6): 324 (3,7): 54
witness diameter X 2.0 · Z 1.0 (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 = 5.099
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 349 nearest-neighbor SWAPs per round in total, at most 8 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(54,6), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt5wtL2_order54_k16.txt, line 1. nonidentity supports a=[2], b=[4, 23].
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

[[108,16,2]] — two-block group-algebra code on SmallGroup(54,6)

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 = 0.59.

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_k16.txt, line 1. This row is SmallGroup(54,6) with nonidentity GAP supports a=[2], b=[4, 23]. 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=16), 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(54,6), 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], b=[4, 23] 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) = 16. 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, 1, 54, 74, 79] [0, 1, 55, 81, 86] [2, 5, 56, 57, 91] [3, 6, 54, 57, 106] [4, 7, 58, 89, 93] [2, 5, 59, 60, 97] [3, 6, 55, 60, 107] [4, 7, 61, 95, 99] [8, 14, 62, 63, 100] [9, 15, 56, 63, 93] [10, 16, 64, 66, 102] [11, 17, 57, 62, 65] [12, 18, 58, 66, 88] [13, 19, 65, 67, 101] [8, 14, 68, 69, 103] [9, 15, 59, 69, 99] [10, 16, 70, 72, 105] [11, 17, 60, 68, 71] [12, 18, 61, 72, 94] [13, 19, 71, 73, 104] [20, 27, 62, 74, 102] [21, 28, 75, 77, 106] [22, 29, 54, 63, 76] [23, 30, 64, 65, 77] [24, 31, 76, 78, 80] [25, 32, 66, 75, 79] [26, 33, 67, 80, 100] [20, 27, 68, 81, 105] [21, 28, 82, 84, 107] [22, 29, 55, 69, 83] [23, 30, 70, 71, 84] [24, 31, 83, 85, 87] [25, 32, 72, 82, 86] [26, 33, 73, 87, 103] [34, 40, 56, 74, 88] [35, 41, 75, 76, 89] [36, 42, 88, 90, 92] [37, 43, 58, 77, 91] [38, 44, 78, 79, 92] [39, 45, 80, 90, 93] [34, 40, 59, 81, 94] [35, 41, 82, 83, 95] [36, 42, 94, 96, 98] [37, 43, 61, 84, 97] [38, 44, 85, 86, 98] [39, 45, 87, 96, 99] [46, 49, 64, 89, 100] [47, 50, 90, 91, 101] [48, 51, 67, 92, 102] [46, 49, 70, 95, 103] [47, 50, 96, 97, 104] [48, 51, 73, 98, 105] [52, 53, 78, 101, 106] [52, 53, 85, 104, 107]
H_Z (54 checks, sparse supports)
[0, 3, 22, 54, 55] [1, 6, 29, 54, 55] [2, 9, 34, 56, 59] [2, 3, 11, 57, 60] [4, 12, 37, 58, 61] [5, 15, 40, 56, 59] [5, 6, 17, 57, 60] [7, 18, 43, 58, 61] [8, 11, 20, 62, 68] [8, 9, 22, 63, 69] [10, 23, 46, 64, 70] [11, 13, 23, 65, 71] [10, 12, 25, 66, 72] [13, 26, 48, 67, 73] [14, 17, 27, 62, 68] [14, 15, 29, 63, 69] [16, 30, 49, 64, 70] [17, 19, 30, 65, 71] [16, 18, 32, 66, 72] [19, 33, 51, 67, 73] [0, 20, 34, 74, 81] [21, 25, 35, 75, 82] [22, 24, 35, 76, 83] [21, 23, 37, 77, 84] [24, 38, 52, 78, 85] [0, 25, 38, 79, 86] [24, 26, 39, 80, 87] [1, 27, 40, 74, 81] [28, 32, 41, 75, 82] [29, 31, 41, 76, 83] [28, 30, 43, 77, 84] [31, 44, 53, 78, 85] [1, 32, 44, 79, 86] [31, 33, 45, 80, 87] [12, 34, 36, 88, 94] [4, 35, 46, 89, 95] [36, 39, 47, 90, 96] [2, 37, 47, 91, 97] [36, 38, 48, 92, 98] [4, 9, 39, 93, 99] [18, 40, 42, 88, 94] [7, 41, 49, 89, 95] [42, 45, 50, 90, 96] [5, 43, 50, 91, 97] [42, 44, 51, 92, 98] [7, 15, 45, 93, 99] [8, 26, 46, 100, 103] [13, 47, 52, 101, 104] [10, 20, 48, 102, 105] [14, 33, 49, 100, 103] [19, 50, 53, 101, 104] [16, 27, 51, 102, 105] [3, 21, 52, 106, 107] [6, 28, 53, 106, 107]
Code ID 108-16-2 · download JSON · raw on GitHub