[[42,16,2]] · QEC Challenge

[[42,16,2]] d =

kd²/n

1.524

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)

[31, 33]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[17, 20]

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 3–4 (mean 3.5) · H_Z 3–4 (mean 3.5)

trapping sets H_X (1,3)×21 (2,0)×21 (3,0)×21 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,3): 21 (1,4): 21 (2,0): 21 (2,4): 63 (2,5): 252 (3,0): 21 (3,3): 7 (3,4): 336 (3,5): 756 (3,6): 1134 (3,7): 1386

trapping sets H_Z (1,3)×21 (2,0)×21 (3,0)×21 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,3): 21 (1,4): 21 (2,0): 21 (2,4): 63 (2,5): 252 (3,0): 21 (3,3): 7 (3,4): 336 (3,5): 756 (3,6): 1134 (3,7): 1386

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(21,2), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt7wtL3_order21_k16.txt, line 1. nonidentity supports a=[2, 4], b=[3, 6, 12].

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

[[42,16,2]] — two-block group-algebra code on SmallGroup(21,2)

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 = 1.52.

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_wt7wtL3_order21_k16.txt, line 1. This row is SmallGroup(21,2) with nonidentity GAP supports a=[2, 4], b=[3, 6, 12]. 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(21,2), 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, 4], b=[3, 6, 12] 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 = 42 - 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 21 (max weight 7) · Z-checks 21 (max weight 7)

H_X (21 checks, sparse supports)

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

H_Z (21 checks, sparse supports)

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

Code ID 42-16-2 · download JSON · raw on GitHub