[[80,20,5]] · QEC Challenge

[[80,20,5]] d =

kd²/n

6.25

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 5, d_Z = 5 · 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 5 · witness weight 5 (claimed upper_bound)

witness operator (support, 5 qubits)

[52, 60, 68, 75, 79]

d_Z 5 · witness weight 5 (claimed upper_bound)

witness operator (support, 5 qubits)

[19, 25, 31, 44, 67]

certificate exact, d = 5 · scipy/HiGHS MILP

X: no logical < 5 exists; Z: no logical < 5 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 (2,4)×160 (3,4)×480 (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 (2,4): 160 (2,6): 800 (3,4): 480 (3,6): 3200 (3,8): 12000 (3,10): 640

trapping sets H_Z (1,4)×80 (2,4)×160 (3,4)×480 (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 (2,4): 160 (2,6): 800 (3,4): 480 (3,6): 3200 (3,8): 12000 (3,10): 640

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_k20.txt, line 2. nonidentity supports a=[2, 5, 8], b=[6, 12, 22].

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

[[80,20,5]] — 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 = 6.25.

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_k20.txt, line 2. This row is SmallGroup(40,1) with nonidentity GAP supports a=[2, 5, 8], b=[6, 12, 22]. 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=20), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 5, 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, 5, 8], b=[6, 12, 22] 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) = 20. 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, 12, 20, 27, 40, 46, 59, 62] [0, 1, 7, 27, 41, 48, 63, 64] [1, 2, 7, 33, 42, 52, 65, 68] [3, 5, 13, 34, 41, 43, 55, 66] [0, 4, 20, 28, 44, 54, 67, 77] [2, 5, 13, 33, 40, 45, 51, 69] [3, 6, 14, 34, 42, 46, 57, 70] [0, 4, 7, 15, 47, 56, 71, 78] [6, 8, 14, 38, 45, 48, 61, 72] [2, 7, 9, 15, 49, 60, 73, 79] [3, 10, 13, 21, 47, 50, 71, 74] [4, 11, 28, 35, 40, 46, 51, 70] [8, 12, 20, 38, 43, 52, 58, 75] [2, 9, 13, 21, 44, 53, 67, 76] [3, 10, 14, 22, 49, 54, 73, 77] [4, 11, 15, 23, 41, 48, 55, 72] [8, 14, 16, 22, 53, 56, 76, 78] [9, 15, 17, 23, 42, 52, 57, 75] [10, 18, 21, 29, 41, 43, 58, 63] [11, 19, 35, 39, 44, 59, 62, 77] [8, 16, 20, 28, 50, 60, 74, 79] [9, 17, 21, 29, 40, 45, 59, 61] [10, 18, 22, 30, 42, 46, 62, 65] [11, 19, 23, 31, 47, 63, 64, 78] [16, 22, 24, 30, 45, 48, 64, 69] [17, 23, 25, 31, 49, 65, 68, 79] [18, 26, 29, 36, 50, 55, 66, 71] [12, 19, 27, 39, 51, 54, 67, 70] [16, 24, 28, 35, 43, 52, 66, 68] [17, 25, 29, 36, 51, 53, 67, 69] [18, 26, 30, 37, 54, 57, 70, 73] [1, 19, 27, 31, 55, 56, 71, 72] [24, 30, 32, 37, 56, 61, 72, 76] [1, 25, 31, 33, 57, 60, 73, 75] [5, 26, 34, 36, 47, 58, 63, 74] [24, 32, 35, 39, 58, 60, 74, 75] [5, 25, 33, 36, 44, 59, 61, 76] [6, 26, 34, 37, 49, 62, 65, 77] [6, 32, 37, 38, 53, 64, 69, 78] [12, 32, 38, 39, 50, 66, 68, 79]

H_Z (40 checks, sparse supports)

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

Code ID 80-20-5 · download JSON · raw on GitHub