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[[108,54,2]] d =
n
108
k
54
d
2
kd²/n
2.0
w
8
X/Z
1
g
0.003
r
5.099
layers
2
swaps
346

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Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · w_X = 8, w_Z = 8 (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)
[35, 41]
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 8 · H_Z 8
qubit degrees H_X 2–6 (mean 4.0) · H_Z 2–6 (mean 4.0)
trapping sets H_X (1,2)×54 (2,0)×54 (3,2)×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 (1,6): 54 (2,0): 54 (2,4): 324 (2,8): 324 (3,2): 810 (3,6): 3294 (3,10): 2592 (3,14): 216
trapping sets H_Z (1,2)×54 (2,0)×54 (3,2)×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 (1,6): 54 (2,0): 54 (2,4): 324 (2,8): 324 (3,2): 810 (3,6): 3294 (3,10): 2592 (3,14): 216
witness diameter X 1.0 · Z 1.4142 (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 346 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(54,10), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL2_order54_k54.txt, line 1. nonidentity supports a=[2], b=[2, 3, 4, 6, 7].
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 ≤ 8 (computed)

How this code was found

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

[[108,54,2]] — two-block group-algebra code on SmallGroup(54,10)

Direction & hypothesis

Target: unrestricted weight-8 frontier for hackathon issue #1155. Reconstruct a published weight-8 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_wt8wtL2_order54_k54.txt, line 1. This row is SmallGroup(54,10) with nonidentity GAP supports a=[2], b=[2, 3, 4, 6, 7]. 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=54), 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 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(54,10), 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=[2, 3, 4, 6, 7] 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) = 54. 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 8) · Z-checks 54 (max weight 8)
H_X (54 checks, sparse supports)
[0, 1, 54, 55, 62, 65, 68, 71] [0, 1, 54, 55, 62, 65, 68, 71] [2, 5, 54, 55, 56, 59, 76, 83] [3, 6, 54, 55, 57, 60, 101, 104] [4, 7, 58, 61, 75, 79, 82, 86] [2, 5, 54, 55, 56, 59, 76, 83] [3, 6, 54, 55, 57, 60, 101, 104] [4, 7, 58, 61, 75, 79, 82, 86] [8, 14, 56, 59, 62, 68, 88, 94] [9, 15, 56, 59, 63, 69, 80, 87] [10, 16, 58, 61, 64, 70, 91, 97] [11, 17, 57, 60, 65, 71, 100, 103] [12, 18, 58, 61, 66, 72, 74, 81] [13, 19, 67, 73, 90, 93, 96, 99] [8, 14, 56, 59, 62, 68, 88, 94] [9, 15, 56, 59, 63, 69, 80, 87] [10, 16, 58, 61, 64, 70, 91, 97] [11, 17, 57, 60, 65, 71, 100, 103] [12, 18, 58, 61, 66, 72, 74, 81] [13, 19, 67, 73, 90, 93, 96, 99] [20, 27, 62, 68, 74, 81, 92, 98] [21, 28, 64, 70, 75, 82, 100, 103] [22, 29, 63, 69, 76, 79, 83, 86] [23, 30, 57, 60, 64, 70, 77, 84] [24, 31, 67, 73, 78, 85, 102, 105] [25, 32, 66, 72, 79, 86, 106, 107] [26, 33, 67, 73, 80, 87, 89, 95] [20, 27, 62, 68, 74, 81, 92, 98] [21, 28, 64, 70, 75, 82, 100, 103] [22, 29, 63, 69, 76, 79, 83, 86] [23, 30, 57, 60, 64, 70, 77, 84] [24, 31, 67, 73, 78, 85, 102, 105] [25, 32, 66, 72, 79, 86, 106, 107] [26, 33, 67, 73, 80, 87, 89, 95] [34, 40, 74, 81, 88, 91, 94, 97] [35, 41, 63, 69, 75, 82, 89, 95] [36, 42, 78, 85, 90, 96, 106, 107] [37, 43, 77, 84, 91, 93, 97, 99] [38, 44, 66, 72, 78, 85, 92, 98] [39, 45, 80, 87, 88, 93, 94, 99] [34, 40, 74, 81, 88, 91, 94, 97] [35, 41, 63, 69, 75, 82, 89, 95] [36, 42, 78, 85, 90, 96, 106, 107] [37, 43, 77, 84, 91, 93, 97, 99] [38, 44, 66, 72, 78, 85, 92, 98] [39, 45, 80, 87, 88, 93, 94, 99] [46, 49, 89, 95, 100, 102, 103, 105] [47, 50, 77, 84, 90, 96, 101, 104] [48, 51, 65, 71, 92, 98, 102, 105] [46, 49, 89, 95, 100, 102, 103, 105] [47, 50, 77, 84, 90, 96, 101, 104] [48, 51, 65, 71, 92, 98, 102, 105] [52, 53, 76, 83, 101, 104, 106, 107] [52, 53, 76, 83, 101, 104, 106, 107]
H_Z (54 checks, sparse supports)
[0, 1, 2, 3, 5, 6, 54, 55] [0, 1, 2, 3, 5, 6, 54, 55] [2, 5, 8, 9, 14, 15, 56, 59] [3, 6, 11, 17, 23, 30, 57, 60] [4, 7, 10, 12, 16, 18, 58, 61] [2, 5, 8, 9, 14, 15, 56, 59] [3, 6, 11, 17, 23, 30, 57, 60] [4, 7, 10, 12, 16, 18, 58, 61] [0, 1, 8, 14, 20, 27, 62, 68] [9, 15, 22, 29, 35, 41, 63, 69] [10, 16, 21, 23, 28, 30, 64, 70] [0, 1, 11, 17, 48, 51, 65, 71] [12, 18, 25, 32, 38, 44, 66, 72] [13, 19, 24, 26, 31, 33, 67, 73] [0, 1, 8, 14, 20, 27, 62, 68] [9, 15, 22, 29, 35, 41, 63, 69] [10, 16, 21, 23, 28, 30, 64, 70] [0, 1, 11, 17, 48, 51, 65, 71] [12, 18, 25, 32, 38, 44, 66, 72] [13, 19, 24, 26, 31, 33, 67, 73] [12, 18, 20, 27, 34, 40, 74, 81] [4, 7, 21, 28, 35, 41, 75, 82] [2, 5, 22, 29, 52, 53, 76, 83] [23, 30, 37, 43, 47, 50, 77, 84] [24, 31, 36, 38, 42, 44, 78, 85] [4, 7, 22, 25, 29, 32, 79, 86] [9, 15, 26, 33, 39, 45, 80, 87] [12, 18, 20, 27, 34, 40, 74, 81] [4, 7, 21, 28, 35, 41, 75, 82] [2, 5, 22, 29, 52, 53, 76, 83] [23, 30, 37, 43, 47, 50, 77, 84] [24, 31, 36, 38, 42, 44, 78, 85] [4, 7, 22, 25, 29, 32, 79, 86] [9, 15, 26, 33, 39, 45, 80, 87] [8, 14, 34, 39, 40, 45, 88, 94] [26, 33, 35, 41, 46, 49, 89, 95] [13, 19, 36, 42, 47, 50, 90, 96] [10, 16, 34, 37, 40, 43, 91, 97] [20, 27, 38, 44, 48, 51, 92, 98] [13, 19, 37, 39, 43, 45, 93, 99] [8, 14, 34, 39, 40, 45, 88, 94] [26, 33, 35, 41, 46, 49, 89, 95] [13, 19, 36, 42, 47, 50, 90, 96] [10, 16, 34, 37, 40, 43, 91, 97] [20, 27, 38, 44, 48, 51, 92, 98] [13, 19, 37, 39, 43, 45, 93, 99] [11, 17, 21, 28, 46, 49, 100, 103] [3, 6, 47, 50, 52, 53, 101, 104] [24, 31, 46, 48, 49, 51, 102, 105] [11, 17, 21, 28, 46, 49, 100, 103] [3, 6, 47, 50, 52, 53, 101, 104] [24, 31, 46, 48, 49, 51, 102, 105] [25, 32, 36, 42, 52, 53, 106, 107] [25, 32, 36, 42, 52, 53, 106, 107]
Code ID 108-54-2 · download JSON · raw on GitHub