[[80,22,4]] · QEC Challenge

[[80,22,4]] d =

kd²/n

4.4

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 4, d_Z = 4 · 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 4 · witness weight 4 (claimed upper_bound)

witness operator (support, 4 qubits)

[49, 53, 56, 60]

d_Z 4 · witness weight 4 (claimed upper_bound)

witness operator (support, 4 qubits)

[21, 29, 58, 61]

certificate exact, d = 4 · scipy/HiGHS MILP

X: no logical < 4 exists; Z: no logical < 4 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 8 · H_Z 8

qubit degrees H_X 4 · H_Z 4

trapping sets H_X (1,4)×80 (2,2)×40 (3,4)×1160 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 80 (2,2): 40 (2,4): 280 (2,6): 440 (3,4): 1160 (3,6): 4240 (3,8): 3840

trapping sets H_Z (1,4)×80 (2,2)×40 (3,4)×1160 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 80 (2,2): 40 (2,4): 280 (2,6): 440 (3,4): 1160 (3,6): 4240 (3,8): 3840

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_wt8wtL4_order40_k22.txt, line 1. nonidentity supports a=[2, 3, 6], b=[3, 5, 10].

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

[[80,22,4]] — 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 = 4.40.

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_wt8wtL4_order40_k22.txt, line 1. This row is SmallGroup(40,1) with nonidentity GAP supports a=[2, 3, 6], b=[3, 5, 10]. 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 ≤ 4, 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=[2, 3, 6], b=[3, 5, 10] 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) = 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 40 (max weight 8) · Z-checks 40 (max weight 8)

H_X (40 checks, sparse supports)

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

H_Z (40 checks, sparse supports)

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

Code ID 80-22-4 · download JSON · raw on GitHub