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 6 · H_Z 6
qubit degrees H_X 2–4 (mean 3.0) · H_Z 2–4 (mean 3.0)
trapping sets H_X (1,2)×26 (2,0)×26 (3,0)×104 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 26
(1,4): 26
(2,0): 26
(2,2): 104
(2,4): 52
(3,0): 104
(3,2): 260
(3,4): 208
(3,6): 104
trapping sets H_Z (1,2)×26 (2,0)×26 (3,0)×104 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 26
(1,4): 26
(2,0): 26
(2,2): 104
(2,4): 52
(3,0): 104
(3,2): 260
(3,4): 208
(3,6): 104
witness diameter X 1.4142 · 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)
Construction & provenance
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on SmallGroup(26,2), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt6wtL2_order26_k26.txt, line 1. nonidentity supports a=[2], b=[2, 3, 4].
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 single (computed from the layout)
weight class weight ≤ 6 (computed)
How this code was found
[[52,26,2]] — two-block group-algebra code on SmallGroup(26,2)
Direction & hypothesis
Target: unrestricted weight-6 frontier for hackathon issue #1155. Reconstruct a published weight-6 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, abelian.zip, diswtabelian_wt6wtL2_order26_k26.txt, line 1. This row is SmallGroup(26,2) with nonidentity GAP supports a=[2], b=[2, 3, 4]. 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=26), 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(26,2), 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] 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 = 52 - rank(HX) - rank(HZ) = 26. 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 26 (max weight 6) · Z-checks 26 (max weight 6)
H_X (26 checks, sparse supports)
[0, 1, 26, 27, 50, 51]
[0, 1, 26, 27, 50, 51]
[2, 3, 26, 27, 28, 29]
[2, 3, 26, 27, 28, 29]
[4, 5, 28, 29, 30, 31]
[4, 5, 28, 29, 30, 31]
[6, 7, 30, 31, 32, 33]
[6, 7, 30, 31, 32, 33]
[8, 9, 32, 33, 34, 35]
[8, 9, 32, 33, 34, 35]
[10, 11, 34, 35, 36, 37]
[10, 11, 34, 35, 36, 37]
[12, 13, 36, 37, 38, 39]
[12, 13, 36, 37, 38, 39]
[14, 15, 38, 39, 40, 41]
[14, 15, 38, 39, 40, 41]
[16, 17, 40, 41, 42, 43]
[16, 17, 40, 41, 42, 43]
[18, 19, 42, 43, 44, 45]
[18, 19, 42, 43, 44, 45]
[20, 21, 44, 45, 46, 47]
[20, 21, 44, 45, 46, 47]
[22, 23, 46, 47, 48, 49]
[22, 23, 46, 47, 48, 49]
[24, 25, 48, 49, 50, 51]
[24, 25, 48, 49, 50, 51]
H_Z (26 checks, sparse supports)
[0, 1, 2, 3, 26, 27]
[0, 1, 2, 3, 26, 27]
[2, 3, 4, 5, 28, 29]
[2, 3, 4, 5, 28, 29]
[4, 5, 6, 7, 30, 31]
[4, 5, 6, 7, 30, 31]
[6, 7, 8, 9, 32, 33]
[6, 7, 8, 9, 32, 33]
[8, 9, 10, 11, 34, 35]
[8, 9, 10, 11, 34, 35]
[10, 11, 12, 13, 36, 37]
[10, 11, 12, 13, 36, 37]
[12, 13, 14, 15, 38, 39]
[12, 13, 14, 15, 38, 39]
[14, 15, 16, 17, 40, 41]
[14, 15, 16, 17, 40, 41]
[16, 17, 18, 19, 42, 43]
[16, 17, 18, 19, 42, 43]
[18, 19, 20, 21, 44, 45]
[18, 19, 20, 21, 44, 45]
[20, 21, 22, 23, 46, 47]
[20, 21, 22, 23, 46, 47]
[22, 23, 24, 25, 48, 49]
[22, 23, 24, 25, 48, 49]
[0, 1, 24, 25, 50, 51]
[0, 1, 24, 25, 50, 51]