← back to the board
[[80,42,2]] d =
n
80
k
42
d
2
kd²/n
2.1
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)
[42, 48]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[36, 39]
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)×80 (2,0)×40 (3,4)×3360 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 80 (2,0): 40 (2,4): 480 (3,4): 3360 (3,8): 640
trapping sets H_Z (1,4)×80 (2,0)×40 (3,4)×3360 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 80 (2,0): 40 (2,4): 480 (3,4): 3360 (3,8): 640

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on SmallGroup(40,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL4_order40_k42.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

[[80,42,2]] — two-block group-algebra code on SmallGroup(40,1)

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 = 2.10.

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_order40_k42.txt, line 1. This row is SmallGroup(40,1) 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=42), 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(40,1), 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 = 80 - rank(HX) - rank(HZ) = 42. 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 40 (max weight 8) · Z-checks 40 (max weight 8)
H_X (40 checks, sparse supports)
[0, 3, 5, 12, 40, 43, 67, 74] [0, 1, 3, 6, 41, 46, 71, 77] [1, 2, 6, 8, 42, 48, 73, 78] [0, 3, 5, 12, 40, 43, 67, 74] [4, 10, 13, 20, 40, 43, 44, 50] [2, 5, 8, 12, 45, 52, 76, 79] [0, 1, 3, 6, 41, 46, 71, 77] [4, 7, 10, 14, 41, 46, 47, 54] [1, 2, 6, 8, 42, 48, 73, 78] [7, 9, 14, 16, 42, 48, 49, 56] [4, 10, 13, 20, 40, 43, 44, 50] [11, 18, 21, 28, 44, 50, 51, 58] [2, 5, 8, 12, 45, 52, 76, 79] [9, 13, 16, 20, 45, 52, 53, 60] [4, 7, 10, 14, 41, 46, 47, 54] [11, 15, 18, 22, 47, 54, 55, 62] [7, 9, 14, 16, 42, 48, 49, 56] [15, 17, 22, 24, 49, 56, 57, 64] [11, 18, 21, 28, 44, 50, 51, 58] [19, 26, 29, 35, 51, 58, 59, 66] [9, 13, 16, 20, 45, 52, 53, 60] [17, 21, 24, 28, 53, 60, 61, 68] [11, 15, 18, 22, 47, 54, 55, 62] [19, 23, 26, 30, 55, 62, 63, 70] [15, 17, 22, 24, 49, 56, 57, 64] [23, 25, 30, 32, 57, 64, 65, 72] [19, 26, 29, 35, 51, 58, 59, 66] [27, 34, 36, 39, 59, 66, 67, 74] [17, 21, 24, 28, 53, 60, 61, 68] [25, 29, 32, 35, 61, 68, 69, 75] [19, 23, 26, 30, 55, 62, 63, 70] [27, 31, 34, 37, 63, 70, 71, 77] [23, 25, 30, 32, 57, 64, 65, 72] [31, 33, 37, 38, 65, 72, 73, 78] [27, 34, 36, 39, 59, 66, 67, 74] [25, 29, 32, 35, 61, 68, 69, 75] [33, 36, 38, 39, 69, 75, 76, 79] [27, 31, 34, 37, 63, 70, 71, 77] [31, 33, 37, 38, 65, 72, 73, 78] [33, 36, 38, 39, 69, 75, 76, 79]
H_Z (40 checks, sparse supports)
[0, 3, 4, 10, 40, 41, 43, 46] [1, 6, 7, 14, 41, 42, 46, 48] [2, 8, 9, 16, 42, 45, 48, 52] [0, 3, 4, 10, 40, 41, 43, 46] [4, 10, 11, 18, 44, 47, 50, 54] [5, 12, 13, 20, 40, 43, 45, 52] [1, 6, 7, 14, 41, 42, 46, 48] [7, 14, 15, 22, 47, 49, 54, 56] [2, 8, 9, 16, 42, 45, 48, 52] [9, 16, 17, 24, 49, 53, 56, 60] [4, 10, 11, 18, 44, 47, 50, 54] [11, 18, 19, 26, 51, 55, 58, 62] [5, 12, 13, 20, 40, 43, 45, 52] [13, 20, 21, 28, 44, 50, 53, 60] [7, 14, 15, 22, 47, 49, 54, 56] [15, 22, 23, 30, 55, 57, 62, 64] [9, 16, 17, 24, 49, 53, 56, 60] [17, 24, 25, 32, 57, 61, 64, 68] [11, 18, 19, 26, 51, 55, 58, 62] [19, 26, 27, 34, 59, 63, 66, 70] [13, 20, 21, 28, 44, 50, 53, 60] [21, 28, 29, 35, 51, 58, 61, 68] [15, 22, 23, 30, 55, 57, 62, 64] [23, 30, 31, 37, 63, 65, 70, 72] [17, 24, 25, 32, 57, 61, 64, 68] [25, 32, 33, 38, 65, 69, 72, 75] [19, 26, 27, 34, 59, 63, 66, 70] [0, 3, 27, 34, 67, 71, 74, 77] [21, 28, 29, 35, 51, 58, 61, 68] [29, 35, 36, 39, 59, 66, 69, 75] [23, 30, 31, 37, 63, 65, 70, 72] [1, 6, 31, 37, 71, 73, 77, 78] [25, 32, 33, 38, 65, 69, 72, 75] [2, 8, 33, 38, 73, 76, 78, 79] [0, 3, 27, 34, 67, 71, 74, 77] [29, 35, 36, 39, 59, 66, 69, 75] [5, 12, 36, 39, 67, 74, 76, 79] [1, 6, 31, 37, 71, 73, 77, 78] [2, 8, 33, 38, 73, 76, 78, 79] [5, 12, 36, 39, 67, 74, 76, 79]
Code ID 80-42-2 · download JSON · raw on GitHub