[[80,4,12]] · QEC Challenge

[[80,4,12]] d ≤

kd²/n

7.2

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 12, d_Z ≤ 12 · w_X = 8, w_Z = 8 (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, 4, 44, 46, 49, 51, 52, 54, 58, 67, 72, 79]

d_Z 12 · witness weight 12 (claimed upper_bound)

witness operator (support, 12 qubits)

[7, 12, 29, 37, 40, 49, 52, 65, 67, 69, 76, 79]

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 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 8 · H_Z 8

qubit degrees H_X 2–6 (mean 4.0) · H_Z 2–6 (mean 4.0)

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 (1,6): 40 (2,2): 40 (2,6): 480 (2,8): 120 (2,10): 360 (3,2): 40 (3,4): 40 (3,6): 2840 (3,8): 2760 (3,10): 7920 (3,12): 3720 (3,14): 2440 (3,16): 200

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 (1,6): 40 (2,2): 40 (2,6): 480 (2,8): 120 (2,10): 360 (3,2): 40 (3,4): 40 (3,6): 2840 (3,8): 2760 (3,10): 7920 (3,12): 3720 (3,14): 2440 (3,16): 200

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(40,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL2_order40_k4.txt, line 6. nonidentity supports a=[10], b=[2, 3, 12, 20, 23].

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

[[80,4,12]] — two-block group-algebra code on SmallGroup(40,1)

Direction & hypothesis

Target: unrestricted weight-8 frontier for hackathon issue #1155. Reconstruct a published weight-8 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 7.20.

What was searched

Scraped GitHub for downloadable parity-check matrix artifacts and parameter tables. From github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL2_order40_k4.txt, line 6. This row is SmallGroup(40,1) with nonidentity GAP supports a=[10], b=[2, 3, 12, 20, 23]. 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=4), 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 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(40,1), 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=[10], b=[2, 3, 12, 20, 23] 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) = 4. 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 8) · Z-checks 40 (max weight 8)

H_X (40 checks, sparse supports)

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

H_Z (40 checks, sparse supports)

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

Code ID 80-4-12 · download JSON · raw on GitHub