[[80,2,12]] · QEC Challenge

[[80,2,12]] d ≤

kd²/n

3.6

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 12, d_Z ≤ 12 · 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 12 · witness weight 12 (claimed upper_bound)

witness operator (support, 12 qubits)

[0, 2, 3, 7, 30, 34, 48, 52, 54, 65, 66, 74]

d_Z 12 · witness weight 12 (claimed upper_bound)

witness operator (support, 12 qubits)

[1, 2, 7, 15, 23, 24, 38, 48, 49, 50, 53, 78]

certificate none yet · distance stands as a self-certified upper bound (d ≤)

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 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)×40 (2,2)×40 (3,2)×40 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 40 (1,4): 40 (2,2): 40 (2,4): 320 (2,6): 240 (3,2): 40 (3,4): 1360 (3,6): 3680 (3,8): 1920 (3,10): 160

trapping sets H_Z (1,2)×40 (2,2)×40 (3,2)×40 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 40 (1,4): 40 (2,2): 40 (2,4): 320 (2,6): 240 (3,2): 40 (3,4): 1360 (3,6): 3680 (3,8): 1920 (3,10): 160

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 ≤ 6 (computed)

How this code was found

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

[[80,2,12]] — two-block group-algebra code on SmallGroup(40,2)

Direction & hypothesis

Target: unrestricted weight-6 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 = 2·144/80 = 3.60.

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_order40_k2.txt, line 1. This row is SmallGroup(40,2) with nonidentity GAP supports a=[6], b=[2,7,31]. 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=2), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 12, not an exact certificate. The published distance (d=12) 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(40,2), 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=[6] and b=[2,7,31] 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 = 80 - rank(HX) - rank(HZ) = 2. 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 40 (max weight 6) · Z-checks 40 (max weight 6)

H_X (40 checks, sparse supports)

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

H_Z (40 checks, sparse supports)

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

Code ID 80-2-12 · download JSON · raw on GitHub