Distance
X/Z asymmetry 1 · d_X = 7, d_Z = 7 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[0, 9, 12, 13, 29, 31, 33]
d_Z 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[5, 7, 9, 24, 25, 29, 36]
certificate exact, d = 7 · scipy/HiGHS MILP
X: no logical < 7 exists; Z: no logical < 7 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 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)×22 (2,2)×22 (3,2)×44 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 22
(1,4): 22
(2,2): 22
(2,4): 187
(2,6): 110
(3,2): 44
(3,4): 858
(3,6): 1672
(3,8): 770
(3,10): 44
trapping sets H_Z (1,2)×22 (2,2)×22 (3,2)×44 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 22
(1,4): 22
(2,2): 22
(2,4): 187
(2,6): 110
(3,2): 44
(3,4): 858
(3,6): 1672
(3,8): 770
(3,10): 44
witness diameter X 4.2426 · Z 4.1231 (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
[[44,4,7]] — two-block group-algebra code on SmallGroup(22,2)
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 = 4.45.
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_order22_k4.txt, line 1. This row is SmallGroup(22,2) with nonidentity GAP supports a=[3], b=[2, 5, 14]. 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=4), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 7, not an exact certificate. The published distance (d=7) 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(22,2), 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=[3] and b=[2, 5, 14] 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 = 44 - rank(HX) - rank(HZ) = 4. 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 22 (max weight 6) · Z-checks 22 (max weight 6)
H_X (22 checks, sparse supports)
[0, 20, 22, 23, 33, 40]
[1, 21, 22, 23, 32, 41]
[0, 2, 24, 25, 35, 42]
[1, 3, 24, 25, 34, 43]
[2, 4, 22, 26, 27, 37]
[3, 5, 23, 26, 27, 36]
[4, 6, 24, 28, 29, 39]
[5, 7, 25, 28, 29, 38]
[6, 8, 26, 30, 31, 41]
[7, 9, 27, 30, 31, 40]
[8, 10, 28, 32, 33, 43]
[9, 11, 29, 32, 33, 42]
[10, 12, 23, 30, 34, 35]
[11, 13, 22, 31, 34, 35]
[12, 14, 25, 32, 36, 37]
[13, 15, 24, 33, 36, 37]
[14, 16, 27, 34, 38, 39]
[15, 17, 26, 35, 38, 39]
[16, 18, 29, 36, 40, 41]
[17, 19, 28, 37, 40, 41]
[18, 20, 31, 38, 42, 43]
[19, 21, 30, 39, 42, 43]
H_Z (22 checks, sparse supports)
[0, 1, 4, 13, 22, 24]
[0, 1, 5, 12, 23, 25]
[2, 3, 6, 15, 24, 26]
[2, 3, 7, 14, 25, 27]
[4, 5, 8, 17, 26, 28]
[4, 5, 9, 16, 27, 29]
[6, 7, 10, 19, 28, 30]
[6, 7, 11, 18, 29, 31]
[8, 9, 12, 21, 30, 32]
[8, 9, 13, 20, 31, 33]
[1, 10, 11, 14, 32, 34]
[0, 10, 11, 15, 33, 35]
[3, 12, 13, 16, 34, 36]
[2, 12, 13, 17, 35, 37]
[5, 14, 15, 18, 36, 38]
[4, 14, 15, 19, 37, 39]
[7, 16, 17, 20, 38, 40]
[6, 16, 17, 21, 39, 41]
[0, 9, 18, 19, 40, 42]
[1, 8, 18, 19, 41, 43]
[2, 11, 20, 21, 22, 42]
[3, 10, 20, 21, 23, 43]