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[[112,58,2]] d =
n
112
k
58
d
2
kd²/n
2.071
w
8
X/Z
1

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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)
[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 4 · H_Z 4
trapping sets H_X (1,4)×112 (2,0)×56 (3,4)×4704 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 112 (2,0): 56 (2,4): 672 (3,4): 4704 (3,8): 896
trapping sets H_Z (1,4)×112 (2,0)×56 (3,4)×4704 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 112 (2,0): 56 (2,4): 672 (3,4): 4704 (3,8): 896

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_wt8wtL4_order56_k58.txt, line 1. nonidentity supports a=[2, 4, 7], b=[4, 5, 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

[[112,58,2]] — 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 = 2.07.

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_wt8wtL4_order56_k58.txt, line 1. This row is SmallGroup(56,1) with nonidentity GAP supports a=[2, 4, 7], b=[4, 5, 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=58), 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=[2, 4, 7], b=[4, 5, 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) = 58. 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, 5, 12, 56, 59, 99, 106] [0, 1, 3, 6, 57, 62, 103, 109] [1, 2, 6, 8, 58, 64, 105, 110] [0, 3, 5, 12, 56, 59, 99, 106] [4, 10, 13, 20, 56, 59, 60, 66] [2, 5, 8, 12, 61, 68, 108, 111] [0, 1, 3, 6, 57, 62, 103, 109] [4, 7, 10, 14, 57, 62, 63, 70] [1, 2, 6, 8, 58, 64, 105, 110] [7, 9, 14, 16, 58, 64, 65, 72] [4, 10, 13, 20, 56, 59, 60, 66] [11, 18, 21, 28, 60, 66, 67, 74] [2, 5, 8, 12, 61, 68, 108, 111] [9, 13, 16, 20, 61, 68, 69, 76] [4, 7, 10, 14, 57, 62, 63, 70] [11, 15, 18, 22, 63, 70, 71, 78] [7, 9, 14, 16, 58, 64, 65, 72] [15, 17, 22, 24, 65, 72, 73, 80] [11, 18, 21, 28, 60, 66, 67, 74] [19, 26, 29, 36, 67, 74, 75, 82] [9, 13, 16, 20, 61, 68, 69, 76] [17, 21, 24, 28, 69, 76, 77, 84] [11, 15, 18, 22, 63, 70, 71, 78] [19, 23, 26, 30, 71, 78, 79, 86] [15, 17, 22, 24, 65, 72, 73, 80] [23, 25, 30, 32, 73, 80, 81, 88] [19, 26, 29, 36, 67, 74, 75, 82] [27, 34, 37, 44, 75, 82, 83, 90] [17, 21, 24, 28, 69, 76, 77, 84] [25, 29, 32, 36, 77, 84, 85, 92] [19, 23, 26, 30, 71, 78, 79, 86] [27, 31, 34, 38, 79, 86, 87, 94] [23, 25, 30, 32, 73, 80, 81, 88] [31, 33, 38, 40, 81, 88, 89, 96] [27, 34, 37, 44, 75, 82, 83, 90] [35, 42, 45, 51, 83, 90, 91, 98] [25, 29, 32, 36, 77, 84, 85, 92] [33, 37, 40, 44, 85, 92, 93, 100] [27, 31, 34, 38, 79, 86, 87, 94] [35, 39, 42, 46, 87, 94, 95, 102] [31, 33, 38, 40, 81, 88, 89, 96] [39, 41, 46, 48, 89, 96, 97, 104] [35, 42, 45, 51, 83, 90, 91, 98] [43, 50, 52, 55, 91, 98, 99, 106] [33, 37, 40, 44, 85, 92, 93, 100] [41, 45, 48, 51, 93, 100, 101, 107] [35, 39, 42, 46, 87, 94, 95, 102] [43, 47, 50, 53, 95, 102, 103, 109] [39, 41, 46, 48, 89, 96, 97, 104] [47, 49, 53, 54, 97, 104, 105, 110] [43, 50, 52, 55, 91, 98, 99, 106] [41, 45, 48, 51, 93, 100, 101, 107] [49, 52, 54, 55, 101, 107, 108, 111] [43, 47, 50, 53, 95, 102, 103, 109] [47, 49, 53, 54, 97, 104, 105, 110] [49, 52, 54, 55, 101, 107, 108, 111]
H_Z (56 checks, sparse supports)
[0, 3, 4, 10, 56, 57, 59, 62] [1, 6, 7, 14, 57, 58, 62, 64] [2, 8, 9, 16, 58, 61, 64, 68] [0, 3, 4, 10, 56, 57, 59, 62] [4, 10, 11, 18, 60, 63, 66, 70] [5, 12, 13, 20, 56, 59, 61, 68] [1, 6, 7, 14, 57, 58, 62, 64] [7, 14, 15, 22, 63, 65, 70, 72] [2, 8, 9, 16, 58, 61, 64, 68] [9, 16, 17, 24, 65, 69, 72, 76] [4, 10, 11, 18, 60, 63, 66, 70] [11, 18, 19, 26, 67, 71, 74, 78] [5, 12, 13, 20, 56, 59, 61, 68] [13, 20, 21, 28, 60, 66, 69, 76] [7, 14, 15, 22, 63, 65, 70, 72] [15, 22, 23, 30, 71, 73, 78, 80] [9, 16, 17, 24, 65, 69, 72, 76] [17, 24, 25, 32, 73, 77, 80, 84] [11, 18, 19, 26, 67, 71, 74, 78] [19, 26, 27, 34, 75, 79, 82, 86] [13, 20, 21, 28, 60, 66, 69, 76] [21, 28, 29, 36, 67, 74, 77, 84] [15, 22, 23, 30, 71, 73, 78, 80] [23, 30, 31, 38, 79, 81, 86, 88] [17, 24, 25, 32, 73, 77, 80, 84] [25, 32, 33, 40, 81, 85, 88, 92] [19, 26, 27, 34, 75, 79, 82, 86] [27, 34, 35, 42, 83, 87, 90, 94] [21, 28, 29, 36, 67, 74, 77, 84] [29, 36, 37, 44, 75, 82, 85, 92] [23, 30, 31, 38, 79, 81, 86, 88] [31, 38, 39, 46, 87, 89, 94, 96] [25, 32, 33, 40, 81, 85, 88, 92] [33, 40, 41, 48, 89, 93, 96, 100] [27, 34, 35, 42, 83, 87, 90, 94] [35, 42, 43, 50, 91, 95, 98, 102] [29, 36, 37, 44, 75, 82, 85, 92] [37, 44, 45, 51, 83, 90, 93, 100] [31, 38, 39, 46, 87, 89, 94, 96] [39, 46, 47, 53, 95, 97, 102, 104] [33, 40, 41, 48, 89, 93, 96, 100] [41, 48, 49, 54, 97, 101, 104, 107] [35, 42, 43, 50, 91, 95, 98, 102] [0, 3, 43, 50, 99, 103, 106, 109] [37, 44, 45, 51, 83, 90, 93, 100] [45, 51, 52, 55, 91, 98, 101, 107] [39, 46, 47, 53, 95, 97, 102, 104] [1, 6, 47, 53, 103, 105, 109, 110] [41, 48, 49, 54, 97, 101, 104, 107] [2, 8, 49, 54, 105, 108, 110, 111] [0, 3, 43, 50, 99, 103, 106, 109] [45, 51, 52, 55, 91, 98, 101, 107] [5, 12, 52, 55, 99, 106, 108, 111] [1, 6, 47, 53, 103, 105, 109, 110] [2, 8, 49, 54, 105, 108, 110, 111] [5, 12, 52, 55, 99, 106, 108, 111]
Code ID 112-58-2 · download JSON · raw on GitHub