[[84,28,2]] · QEC Challenge

[[84,28,2]] 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)

[56, 59]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[25, 28]

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)×42 (2,0)×21 (3,2)×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 (1,5): 42 (2,0): 21 (2,3): 84 (2,4): 42 (2,5): 252 (2,6): 21 (2,8): 252 (3,2): 168 (3,3): 504 (3,4): 126 (3,5): 1008 (3,6): 756 (3,7): 126 (3,8): 3318 (3,9): 266 (3,11): 2142

trapping sets H_Z (1,2)×42 (2,0)×21 (3,2)×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 (1,5): 42 (2,0): 21 (2,3): 84 (2,4): 42 (2,5): 252 (2,6): 21 (2,8): 252 (3,2): 168 (3,3): 504 (3,4): 126 (3,5): 1008 (3,6): 756 (3,7): 126 (3,8): 3318 (3,9): 266 (3,11): 2142

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_wt7wtL2_order42_k28.txt, line 1. nonidentity supports a=[2], b=[2, 4, 8, 13].

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

[[84,28,2]] — two-block group-algebra code on SmallGroup(42,6)

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_order42_k28.txt, line 1. This row is SmallGroup(42,6) with nonidentity GAP supports a=[2], b=[2, 4, 8, 13]. 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=28), 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, 4, 8, 13] 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) = 28. 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 7) · Z-checks 42 (max weight 7)

H_X (42 checks, sparse supports)

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

H_Z (42 checks, sparse supports)

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

Code ID 84-28-2 · download JSON · raw on GitHub