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)×24 (2,2)×48 (3,2)×216 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 24
(1,4): 24
(2,2): 48
(2,4): 156
(2,6): 120
(3,2): 216
(3,4): 720
(3,6): 1104
(3,8): 528
(3,10): 48
trapping sets H_Z (1,2)×24 (2,2)×48 (3,2)×216 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 24
(1,4): 24
(2,2): 48
(2,4): 156
(2,6): 120
(3,2): 216
(3,4): 720
(3,6): 1104
(3,8): 528
(3,10): 48
witness diameter X 2.2361 · Z 2.2361 (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)
How this code was found
[[48,12,4]] — two-block group-algebra code on SmallGroup(24,2)
Direction & hypothesis
Target: unrestricted weight-6 frontier for hackathon issue #1155. Reconstruct a published 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 4.00.
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_order24_k12.txt, line 1. This row is SmallGroup(24,2) with nonidentity GAP supports a=[4], b=[4, 6, 15]. 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=12), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 4, not an exact certificate. The published distance (d=4) 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
Union Alpha (maker currently anonymous), 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(24,2), 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=[4] and b=[4, 6, 15] 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 = 48 - rank(HX) - rank(HZ) = 12. 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 24 (max weight 6) · Z-checks 24 (max weight 6)
H_X (24 checks, sparse supports)
[0, 11, 24, 35, 43, 47]
[1, 15, 25, 32, 39, 41]
[2, 18, 26, 30, 39, 42]
[0, 3, 24, 27, 36, 44]
[3, 4, 27, 28, 43, 47]
[5, 21, 24, 28, 29, 45]
[1, 6, 25, 30, 40, 46]
[6, 7, 30, 31, 32, 41]
[8, 22, 32, 37, 45, 46]
[2, 9, 25, 26, 31, 33]
[9, 10, 30, 33, 34, 39]
[4, 11, 28, 35, 36, 44]
[12, 23, 26, 34, 36, 47]
[5, 13, 27, 29, 35, 37]
[13, 14, 24, 28, 37, 38]
[7, 15, 31, 39, 40, 46]
[8, 16, 29, 32, 38, 40]
[16, 17, 37, 40, 41, 45]
[10, 18, 25, 31, 34, 42]
[12, 19, 33, 36, 42, 43]
[19, 20, 26, 34, 43, 44]
[14, 21, 27, 35, 38, 45]
[17, 22, 29, 38, 41, 46]
[20, 23, 33, 42, 44, 47]
H_Z (24 checks, sparse supports)
[0, 3, 5, 14, 24, 27]
[1, 6, 9, 18, 25, 30]
[2, 9, 12, 20, 26, 33]
[3, 4, 13, 21, 27, 28]
[4, 5, 11, 14, 28, 35]
[5, 13, 16, 22, 29, 37]
[2, 6, 7, 10, 30, 31]
[7, 9, 15, 18, 31, 39]
[1, 7, 8, 16, 32, 40]
[9, 10, 19, 23, 33, 34]
[10, 12, 18, 20, 34, 42]
[0, 11, 13, 21, 24, 35]
[3, 11, 12, 19, 36, 43]
[8, 13, 14, 17, 37, 38]
[14, 16, 21, 22, 38, 45]
[1, 2, 10, 15, 25, 39]
[6, 15, 16, 17, 40, 41]
[1, 7, 17, 22, 41, 46]
[2, 18, 19, 23, 26, 42]
[0, 4, 19, 20, 43, 44]
[3, 11, 20, 23, 44, 47]
[5, 8, 17, 21, 29, 45]
[6, 8, 15, 22, 32, 46]
[0, 4, 12, 23, 36, 47]