[[80,24,2]] · QEC Challenge

[[80,24,2]] d =

kd²/n

1.2

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · 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 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[58, 62]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[61, 65]

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 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 (2,0)×40 (3,0)×32 (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 (2,0): 40 (2,2): 32 (2,4): 256 (2,6): 160 (3,0): 32 (3,2): 144 (3,4): 1200 (3,6): 2112 (3,8): 1152

trapping sets H_Z (1,2)×40 (2,0)×20 (3,2)×256 (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 (2,0): 20 (2,2): 32 (2,4): 376 (3,2): 256 (3,4): 1848 (3,6): 912 (3,8): 80

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(40,5), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt6wtL2_order40_k24.txt, line 1. nonidentity supports a=[2], b=[2, 10, 14].

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 ≤ 6 (computed)

How this code was found

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

[[80,24,2]] — two-block group-algebra code on SmallGroup(40,5)

Direction & hypothesis

Target: unrestricted weight-6 frontier for hackathon issue #1155. Reconstruct a published weight-6 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 1.20.

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_wt6wtL2_order40_k24.txt, line 1. This row is SmallGroup(40,5) with nonidentity GAP supports a=[2], b=[2, 10, 14]. 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=24), 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(40,5), 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=[2, 10, 14] 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) = 24. 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, 1, 40, 41, 60, 78] [0, 1, 40, 41, 56, 79] [2, 5, 42, 45, 47, 67] [3, 6, 43, 46, 53, 73] [4, 7, 44, 48, 52, 71] [2, 5, 42, 44, 45, 71] [3, 6, 43, 46, 49, 76] [4, 7, 47, 48, 52, 67] [8, 12, 48, 52, 54, 74] [9, 13, 40, 41, 49, 76] [10, 14, 42, 45, 50, 77] [11, 15, 51, 56, 63, 79] [8, 12, 48, 50, 52, 77] [9, 13, 40, 41, 53, 73] [10, 14, 42, 45, 54, 74] [11, 15, 55, 59, 60, 78] [16, 20, 43, 46, 56, 79] [17, 21, 44, 57, 69, 71] [18, 22, 49, 58, 70, 76] [19, 23, 55, 59, 64, 75] [16, 20, 43, 46, 60, 78] [17, 21, 47, 61, 65, 67] [18, 22, 53, 62, 66, 73] [19, 23, 51, 63, 68, 72] [24, 28, 50, 64, 75, 77] [25, 29, 51, 61, 63, 65] [26, 30, 57, 62, 66, 69] [27, 31, 47, 67, 68, 72] [24, 28, 54, 68, 72, 74] [25, 29, 55, 57, 59, 69] [26, 30, 58, 61, 65, 70] [27, 31, 44, 64, 71, 75] [32, 35, 58, 68, 70, 72] [33, 36, 53, 55, 59, 73] [34, 37, 54, 61, 65, 74] [32, 35, 62, 64, 66, 75] [33, 36, 49, 51, 63, 76] [34, 37, 50, 57, 69, 77] [38, 39, 60, 62, 66, 78] [38, 39, 56, 58, 70, 79]

H_Z (40 checks, sparse supports)

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

Code ID 80-24-2 · download JSON · raw on GitHub