[[84,12,8]] · QEC Challenge

[[84,12,8]] d =

kd²/n

9.143

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 8, d_Z = 8 · 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 8 · witness weight 8 (claimed upper_bound)

witness operator (support, 8 qubits)

[32, 37, 58, 63, 67, 72, 73, 78]

d_Z 8 · witness weight 8 (claimed upper_bound)

witness operator (support, 8 qubits)

[58, 63, 64, 69, 70, 75, 81, 83]

certificate exact, d = 8 · scipy/HiGHS MILP

X: no logical < 8 exists; Z: no logical < 8 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 6 · H_Z 6 (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)×42 (2,4)×126 (3,3)×84 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 42 (1,4): 42 (2,4): 126 (2,5): 504 (2,6): 252 (3,3): 84 (3,4): 126 (3,5): 504 (3,6): 4256 (3,7): 5838 (3,8): 1722 (3,9): 756 (3,10): 168

trapping sets H_Z (1,3)×42 (2,4)×126 (3,3)×84 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 42 (1,4): 42 (2,4): 126 (2,5): 504 (2,6): 252 (3,3): 84 (3,4): 126 (3,5): 504 (3,6): 4256 (3,7): 5838 (3,8): 1722 (3,9): 756 (3,10): 168

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction contributed via qldpc submit

model DeepSeek V4 Flash 0731 (claimed, not verified)

date 2026-09-18

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

[[84,12,8]] — 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 = 9.14.

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_order42_k12.txt, line 1. This row is SmallGroup(42,6) with nonidentity GAP supports a=[4, 18], b=[3, 13, 25]. 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=12), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 8, not an exact certificate. The published distance (d=8) 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

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

H_Z (42 checks, sparse supports)

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

Code ID 84-12-8 · download JSON · raw on GitHub