← back to the board
[[112,56,2]] d =
n
112
k
56
d
2
kd²/n
2.0
w
8
X/Z
1

Share this result

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)
[41, 48]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[9, 16]
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)×56 (2,0)×56 (3,2)×840 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 56 (1,6): 56 (2,0): 56 (2,4): 336 (2,8): 336 (3,2): 840 (3,6): 3416 (3,10): 2688 (3,14): 224
trapping sets H_Z (1,2)×56 (2,0)×56 (3,2)×840 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 56 (1,6): 56 (2,0): 56 (2,4): 336 (2,8): 336 (3,2): 840 (3,6): 3416 (3,10): 2688 (3,14): 224

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on SmallGroup(56,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL2_order56_k56.txt, line 1. nonidentity supports a=[4], b=[2, 4, 5, 7, 11].
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 ≤ 8 (computed)

How this code was found

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

[[112,56,2]] — two-block group-algebra code on SmallGroup(56,1)

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_order56_k56.txt, line 1. This row is SmallGroup(56,1) with nonidentity GAP supports a=[4], b=[2, 4, 5, 7, 11]. 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=56), 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(56,1), 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=[4], b=[2, 4, 5, 7, 11] 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 = 112 - rank(HX) - rank(HZ) = 56. 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 56 (max weight 8) · Z-checks 56 (max weight 8)
H_X (56 checks, sparse supports)
[0, 3, 56, 59, 61, 68, 99, 106] [1, 6, 56, 57, 59, 62, 103, 109] [2, 8, 57, 58, 62, 64, 105, 110] [0, 3, 56, 59, 61, 68, 99, 106] [4, 10, 56, 59, 60, 66, 108, 111] [5, 12, 58, 61, 64, 68, 108, 111] [1, 6, 56, 57, 59, 62, 103, 109] [7, 14, 57, 62, 63, 70, 99, 106] [2, 8, 57, 58, 62, 64, 105, 110] [9, 16, 58, 64, 65, 72, 103, 109] [4, 10, 56, 59, 60, 66, 108, 111] [11, 18, 60, 66, 67, 74, 101, 107] [5, 12, 58, 61, 64, 68, 108, 111] [13, 20, 61, 68, 69, 76, 105, 110] [7, 14, 57, 62, 63, 70, 99, 106] [15, 22, 63, 70, 71, 78, 91, 98] [9, 16, 58, 64, 65, 72, 103, 109] [17, 24, 65, 72, 73, 80, 95, 102] [11, 18, 60, 66, 67, 74, 101, 107] [19, 26, 67, 74, 75, 82, 93, 100] [13, 20, 61, 68, 69, 76, 105, 110] [21, 28, 69, 76, 77, 84, 97, 104] [15, 22, 63, 70, 71, 78, 91, 98] [23, 30, 71, 78, 79, 83, 86, 90] [17, 24, 65, 72, 73, 80, 95, 102] [25, 32, 73, 80, 81, 87, 88, 94] [19, 26, 67, 74, 75, 82, 93, 100] [27, 34, 75, 82, 83, 85, 90, 92] [21, 28, 69, 76, 77, 84, 97, 104] [29, 36, 77, 84, 85, 89, 92, 96] [23, 30, 71, 78, 79, 83, 86, 90] [31, 38, 75, 79, 82, 86, 87, 94] [25, 32, 73, 80, 81, 87, 88, 94] [33, 40, 79, 81, 86, 88, 89, 96] [27, 34, 75, 82, 83, 85, 90, 92] [35, 42, 77, 83, 84, 90, 91, 98] [29, 36, 77, 84, 85, 89, 92, 96] [37, 44, 81, 85, 88, 92, 93, 100] [31, 38, 75, 79, 82, 86, 87, 94] [39, 46, 67, 74, 87, 94, 95, 102] [33, 40, 79, 81, 86, 88, 89, 96] [41, 48, 71, 78, 89, 96, 97, 104] [35, 42, 77, 83, 84, 90, 91, 98] [43, 50, 69, 76, 91, 98, 99, 106] [37, 44, 81, 85, 88, 92, 93, 100] [45, 51, 73, 80, 93, 100, 101, 107] [39, 46, 67, 74, 87, 94, 95, 102] [47, 53, 60, 66, 95, 102, 103, 109] [41, 48, 71, 78, 89, 96, 97, 104] [49, 54, 63, 70, 97, 104, 105, 110] [43, 50, 69, 76, 91, 98, 99, 106] [45, 51, 73, 80, 93, 100, 101, 107] [52, 55, 65, 72, 101, 107, 108, 111] [47, 53, 60, 66, 95, 102, 103, 109] [49, 54, 63, 70, 97, 104, 105, 110] [52, 55, 65, 72, 101, 107, 108, 111]
H_Z (56 checks, sparse supports)
[0, 1, 3, 4, 6, 10, 56, 59] [1, 2, 6, 7, 8, 14, 57, 62] [2, 5, 8, 9, 12, 16, 58, 64] [0, 1, 3, 4, 6, 10, 56, 59] [4, 10, 11, 18, 47, 53, 60, 66] [0, 3, 5, 12, 13, 20, 61, 68] [1, 2, 6, 7, 8, 14, 57, 62] [7, 14, 15, 22, 49, 54, 63, 70] [2, 5, 8, 9, 12, 16, 58, 64] [9, 16, 17, 24, 52, 55, 65, 72] [4, 10, 11, 18, 47, 53, 60, 66] [11, 18, 19, 26, 39, 46, 67, 74] [0, 3, 5, 12, 13, 20, 61, 68] [13, 20, 21, 28, 43, 50, 69, 76] [7, 14, 15, 22, 49, 54, 63, 70] [15, 22, 23, 30, 41, 48, 71, 78] [9, 16, 17, 24, 52, 55, 65, 72] [17, 24, 25, 32, 45, 51, 73, 80] [11, 18, 19, 26, 39, 46, 67, 74] [19, 26, 27, 31, 34, 38, 75, 82] [13, 20, 21, 28, 43, 50, 69, 76] [21, 28, 29, 35, 36, 42, 77, 84] [15, 22, 23, 30, 41, 48, 71, 78] [23, 30, 31, 33, 38, 40, 79, 86] [17, 24, 25, 32, 45, 51, 73, 80] [25, 32, 33, 37, 40, 44, 81, 88] [19, 26, 27, 31, 34, 38, 75, 82] [23, 27, 30, 34, 35, 42, 83, 90] [21, 28, 29, 35, 36, 42, 77, 84] [27, 29, 34, 36, 37, 44, 85, 92] [23, 30, 31, 33, 38, 40, 79, 86] [25, 31, 32, 38, 39, 46, 87, 94] [25, 32, 33, 37, 40, 44, 81, 88] [29, 33, 36, 40, 41, 48, 89, 96] [23, 27, 30, 34, 35, 42, 83, 90] [15, 22, 35, 42, 43, 50, 91, 98] [27, 29, 34, 36, 37, 44, 85, 92] [19, 26, 37, 44, 45, 51, 93, 100] [25, 31, 32, 38, 39, 46, 87, 94] [17, 24, 39, 46, 47, 53, 95, 102] [29, 33, 36, 40, 41, 48, 89, 96] [21, 28, 41, 48, 49, 54, 97, 104] [15, 22, 35, 42, 43, 50, 91, 98] [0, 3, 7, 14, 43, 50, 99, 106] [19, 26, 37, 44, 45, 51, 93, 100] [11, 18, 45, 51, 52, 55, 101, 107] [17, 24, 39, 46, 47, 53, 95, 102] [1, 6, 9, 16, 47, 53, 103, 109] [21, 28, 41, 48, 49, 54, 97, 104] [2, 8, 13, 20, 49, 54, 105, 110] [0, 3, 7, 14, 43, 50, 99, 106] [11, 18, 45, 51, 52, 55, 101, 107] [4, 5, 10, 12, 52, 55, 108, 111] [1, 6, 9, 16, 47, 53, 103, 109] [2, 8, 13, 20, 49, 54, 105, 110] [4, 5, 10, 12, 52, 55, 108, 111]
Code ID 112-56-2 · download JSON · raw on GitHub