[[42,18,3]] · QEC Challenge

[[42,18,3]] d =

kd²/n

3.857

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 3 · witness weight 3 (claimed upper_bound)

witness operator (support, 3 qubits)

[29, 31, 33]

d_Z 3 · witness weight 3 (claimed upper_bound)

witness operator (support, 3 qubits)

[14, 16, 18]

certificate exact, d = 3 · scipy/HiGHS MILP

X: no logical < 3 exists; Z: no logical < 3 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 8 · H_Z 8

qubit degrees H_X 4 · H_Z 4

trapping sets H_X (1,4)×42 (2,4)×168 (3,0)×35 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 42 (2,4): 168 (2,6): 252 (3,0): 35 (3,4): 693 (3,6): 1974 (3,8): 1848

trapping sets H_Z (1,4)×42 (2,4)×168 (3,0)×35 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 42 (2,4): 168 (2,6): 252 (3,0): 35 (3,4): 693 (3,6): 1974 (3,8): 1848

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_wt8wtL4_order21_k18.txt, line 1. nonidentity supports a=[2, 3, 5], b=[2, 6, 8].

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

[[42,18,3]] — two-block group-algebra code on SmallGroup(21,2)

Direction & hypothesis

Target: unrestricted weight-8 frontier for hackathon issue #1155. Reconstruct a published weight-8 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 3.86.

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_wt8wtL4_order21_k18.txt, line 1. This row is SmallGroup(21,2) with nonidentity GAP supports a=[2, 3, 5], b=[2, 6, 8]. 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=18), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 3, not an exact certificate. The published distance 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, 3, 5], b=[2, 6, 8] 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) = 18. 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 8) · Z-checks 21 (max weight 8)

H_X (21 checks, sparse supports)

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

H_Z (21 checks, sparse supports)

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

Code ID 42-18-3 · download JSON · raw on GitHub