[[48,16,2]] · QEC Challenge

[[48,16,2]] d =

kd²/n

1.333

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · w_X = 7, w_Z = 7 (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)

[24, 28]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[6, 15]

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 7 · H_Z 7

qubit degrees H_X 2–5 (mean 3.5) · H_Z 2–5 (mean 3.5)

trapping sets H_X (1,2)×24 (2,0)×12 (3,2)×96 (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 (1,5): 24 (2,0): 12 (2,3): 48 (2,4): 24 (2,5): 144 (2,6): 12 (2,8): 144 (3,2): 96 (3,3): 288 (3,4): 72 (3,5): 576 (3,6): 432 (3,7): 72 (3,8): 1896 (3,9): 152 (3,11): 1224

trapping sets H_Z (1,2)×24 (2,0)×12 (3,2)×96 (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 (1,5): 24 (2,0): 12 (2,3): 48 (2,4): 24 (2,5): 144 (2,6): 12 (2,8): 144 (3,2): 96 (3,3): 288 (3,4): 72 (3,5): 576 (3,6): 432 (3,7): 72 (3,8): 1896 (3,9): 152 (3,11): 1224

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(24,2), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt7wtL2_order24_k16.txt, line 1. nonidentity supports a=[5], b=[2, 3, 8, 9].

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

[[48,16,2]] — two-block group-algebra code on SmallGroup(24,2)

Direction & hypothesis

Target: unrestricted weight-8 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 = 1.33.

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_wt7wtL2_order24_k16.txt, line 1. This row is SmallGroup(24,2) with nonidentity GAP supports a=[5], b=[2, 3, 8, 9]. 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=16), 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(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=[5], b=[2, 3, 8, 9] 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) = 16. 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 7) · Z-checks 24 (max weight 7)

H_X (24 checks, sparse supports)

[0, 4, 24, 26, 30, 32, 39] [1, 7, 24, 25, 28, 29, 36] [2, 10, 24, 26, 32, 37, 45] [3, 11, 25, 27, 31, 33, 40] [0, 4, 28, 30, 34, 39, 41] [5, 14, 25, 26, 29, 34, 36] [6, 15, 27, 30, 35, 37, 43] [1, 7, 24, 28, 31, 38, 44] [8, 17, 24, 26, 32, 43, 47] [9, 18, 27, 29, 33, 38, 40] [2, 10, 28, 34, 37, 41, 45] [3, 11, 25, 31, 35, 42, 46] [12, 20, 25, 29, 32, 36, 41] [13, 21, 30, 33, 37, 42, 43] [5, 14, 26, 31, 34, 38, 44] [6, 15, 27, 35, 39, 45, 47] [16, 22, 27, 33, 36, 40, 44] [8, 17, 28, 34, 41, 43, 47] [9, 18, 29, 35, 38, 42, 46] [19, 23, 30, 37, 40, 43, 46] [12, 20, 31, 32, 38, 41, 44] [13, 21, 33, 39, 42, 45, 47] [16, 22, 35, 36, 42, 44, 46] [19, 23, 39, 40, 45, 46, 47]

H_Z (24 checks, sparse supports)

[0, 1, 2, 7, 8, 24, 28] [1, 3, 5, 11, 12, 25, 31] [0, 2, 5, 8, 14, 26, 34] [3, 6, 9, 15, 16, 27, 35] [1, 4, 7, 10, 17, 24, 28] [1, 5, 9, 12, 18, 29, 38] [0, 4, 6, 13, 19, 30, 39] [3, 7, 11, 14, 20, 25, 31] [0, 2, 8, 12, 20, 32, 41] [3, 9, 13, 16, 21, 33, 42] [4, 5, 10, 14, 17, 26, 34] [6, 11, 15, 18, 22, 27, 35] [1, 5, 12, 16, 22, 36, 44] [2, 6, 10, 13, 19, 37, 45] [7, 9, 14, 18, 20, 29, 38] [0, 4, 15, 21, 23, 30, 39] [3, 9, 16, 19, 23, 40, 46] [4, 10, 12, 17, 20, 32, 41] [11, 13, 18, 21, 22, 33, 42] [6, 8, 13, 17, 19, 43, 47] [7, 14, 16, 20, 22, 36, 44] [2, 10, 15, 21, 23, 37, 45] [11, 18, 19, 22, 23, 40, 46] [8, 15, 17, 21, 23, 43, 47]

Code ID 48-16-2 · download JSON · raw on GitHub