[[84,42,2]] · QEC Challenge

[[84,42,2]] d =

kd²/n

2.0

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · 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 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[19, 22]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[44, 46]

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

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

(1,2): 42 (1,4): 42 (2,0): 42 (2,2): 168 (2,4): 84 (3,0): 168 (3,2): 420 (3,4): 336 (3,6): 168

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

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

(1,2): 42 (1,4): 42 (2,0): 42 (2,2): 168 (2,4): 84 (3,0): 168 (3,2): 420 (3,4): 336 (3,6): 168

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(42,6), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt6wtL2_order42_k42.txt, line 1. nonidentity supports a=[2], b=[2, 8, 11].

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 ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[84,42,2]] — two-block group-algebra code on SmallGroup(42,6)

Direction & hypothesis

Target: unrestricted weight-6 frontier for hackathon issue #1155. Reconstruct a published weight-6 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 2.00.

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_order42_k42.txt, line 1. This row is SmallGroup(42,6) with nonidentity GAP supports a=[2], b=[2, 8, 11]. 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=42), 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(42,6), 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], b=[2, 8, 11] 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 = 84 - rank(HX) - rank(HZ) = 42. 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 42 (max weight 6) · Z-checks 42 (max weight 6)

H_X (42 checks, sparse supports)

[0, 1, 42, 43, 82, 83] [0, 1, 42, 43, 82, 83] [2, 4, 44, 46, 74, 77] [3, 5, 45, 47, 48, 51] [2, 4, 44, 46, 74, 77] [3, 5, 45, 47, 48, 51] [6, 9, 48, 51, 79, 81] [7, 10, 42, 43, 49, 52] [8, 11, 50, 53, 54, 57] [6, 9, 48, 51, 79, 81] [7, 10, 42, 43, 49, 52] [8, 11, 50, 53, 54, 57] [12, 15, 44, 46, 54, 57] [13, 16, 45, 47, 55, 58] [14, 17, 56, 59, 60, 63] [12, 15, 44, 46, 54, 57] [13, 16, 45, 47, 55, 58] [14, 17, 56, 59, 60, 63] [18, 21, 49, 52, 60, 63] [19, 22, 50, 53, 61, 64] [20, 23, 62, 65, 66, 69] [18, 21, 49, 52, 60, 63] [19, 22, 50, 53, 61, 64] [20, 23, 62, 65, 66, 69] [24, 27, 55, 58, 66, 69] [25, 28, 56, 59, 67, 70] [26, 29, 68, 71, 72, 75] [24, 27, 55, 58, 66, 69] [25, 28, 56, 59, 67, 70] [26, 29, 68, 71, 72, 75] [30, 33, 61, 64, 72, 75] [31, 34, 62, 65, 73, 76] [32, 35, 74, 77, 78, 80] [30, 33, 61, 64, 72, 75] [31, 34, 62, 65, 73, 76] [32, 35, 74, 77, 78, 80] [36, 38, 67, 70, 78, 80] [37, 39, 68, 71, 79, 81] [36, 38, 67, 70, 78, 80] [37, 39, 68, 71, 79, 81] [40, 41, 73, 76, 82, 83] [40, 41, 73, 76, 82, 83]

H_Z (42 checks, sparse supports)

[0, 1, 7, 10, 42, 43] [0, 1, 7, 10, 42, 43] [2, 4, 12, 15, 44, 46] [3, 5, 13, 16, 45, 47] [2, 4, 12, 15, 44, 46] [3, 5, 13, 16, 45, 47] [3, 5, 6, 9, 48, 51] [7, 10, 18, 21, 49, 52] [8, 11, 19, 22, 50, 53] [3, 5, 6, 9, 48, 51] [7, 10, 18, 21, 49, 52] [8, 11, 19, 22, 50, 53] [8, 11, 12, 15, 54, 57] [13, 16, 24, 27, 55, 58] [14, 17, 25, 28, 56, 59] [8, 11, 12, 15, 54, 57] [13, 16, 24, 27, 55, 58] [14, 17, 25, 28, 56, 59] [14, 17, 18, 21, 60, 63] [19, 22, 30, 33, 61, 64] [20, 23, 31, 34, 62, 65] [14, 17, 18, 21, 60, 63] [19, 22, 30, 33, 61, 64] [20, 23, 31, 34, 62, 65] [20, 23, 24, 27, 66, 69] [25, 28, 36, 38, 67, 70] [26, 29, 37, 39, 68, 71] [20, 23, 24, 27, 66, 69] [25, 28, 36, 38, 67, 70] [26, 29, 37, 39, 68, 71] [26, 29, 30, 33, 72, 75] [31, 34, 40, 41, 73, 76] [2, 4, 32, 35, 74, 77] [26, 29, 30, 33, 72, 75] [31, 34, 40, 41, 73, 76] [2, 4, 32, 35, 74, 77] [32, 35, 36, 38, 78, 80] [6, 9, 37, 39, 79, 81] [32, 35, 36, 38, 78, 80] [6, 9, 37, 39, 79, 81] [0, 1, 40, 41, 82, 83] [0, 1, 40, 41, 82, 83]

Code ID 84-42-2 · download JSON · raw on GitHub