[[84,22,2]] · QEC Challenge

[[84,22,2]] d =

kd²/n

1.048

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)

[24, 27]

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 (3,5)×1890 (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 (2,5): 420 (2,8): 420 (3,5): 1890 (3,8): 7770 (3,9): 2240 (3,10): 840 (3,13): 420

trapping sets H_Z (1,2)×42 (2,0)×21 (3,5)×1890 (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 (2,5): 420 (2,8): 420 (3,5): 1890 (3,8): 7770 (3,9): 2240 (3,10): 840 (3,13): 420

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_k22.txt, line 1. nonidentity supports a=[2], b=[3, 13, 19, 31].

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

[[84,22,2]] — 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 = 1.05.

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_k22.txt, line 1. This row is SmallGroup(42,6) with nonidentity GAP supports a=[2], b=[3, 13, 19, 31]. 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=22), 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=[3, 13, 19, 31] 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) = 22. 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, 48, 61, 73, 79] [0, 1, 43, 51, 64, 76, 81] [2, 4, 42, 44, 66, 78, 82] [3, 5, 44, 45, 54, 67, 79] [2, 4, 43, 46, 69, 80, 83] [3, 5, 46, 47, 57, 70, 81] [6, 9, 44, 48, 56, 68, 74] [7, 10, 45, 48, 49, 72, 82] [8, 11, 44, 49, 50, 60, 73] [6, 9, 46, 51, 59, 71, 77] [7, 10, 47, 51, 52, 75, 83] [8, 11, 46, 52, 53, 63, 76] [12, 15, 42, 49, 54, 62, 74] [13, 16, 48, 50, 54, 55, 78] [14, 17, 49, 55, 56, 66, 79] [12, 15, 43, 52, 57, 65, 77] [13, 16, 51, 53, 57, 58, 80] [14, 17, 52, 58, 59, 69, 81] [18, 21, 42, 45, 55, 60, 68] [19, 22, 54, 56, 60, 61, 82] [20, 23, 44, 55, 61, 62, 72] [18, 21, 43, 47, 58, 63, 71] [19, 22, 57, 59, 63, 64, 83] [20, 23, 46, 58, 64, 65, 75] [24, 27, 45, 50, 61, 66, 74] [25, 28, 48, 60, 62, 66, 67] [26, 29, 49, 61, 67, 68, 78] [24, 27, 47, 53, 64, 69, 77] [25, 28, 51, 63, 65, 69, 70] [26, 29, 52, 64, 70, 71, 80] [30, 33, 42, 50, 56, 67, 72] [31, 34, 54, 66, 68, 72, 73] [32, 35, 55, 67, 73, 74, 82] [30, 33, 43, 53, 59, 70, 75] [31, 34, 57, 69, 71, 75, 76] [32, 35, 58, 70, 76, 77, 83] [36, 38, 45, 56, 62, 73, 78] [37, 39, 60, 72, 74, 78, 79] [36, 38, 47, 59, 65, 76, 80] [37, 39, 63, 75, 77, 80, 81] [40, 41, 50, 62, 68, 79, 82] [40, 41, 53, 65, 71, 81, 83]

H_Z (42 checks, sparse supports)

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

Code ID 84-22-2 · download JSON · raw on GitHub