[[80,40,2]] · QEC Challenge

[[80,40,2]] d =

kd²/n

2.0

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)

[43, 51]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[3, 11]

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)×160 (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): 160 (2,4): 80 (3,0): 160 (3,2): 400 (3,4): 320 (3,6): 160

trapping sets H_Z (1,2)×40 (2,0)×40 (3,0)×160 (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): 40 (2,2): 160 (2,4): 80 (3,0): 160 (3,2): 400 (3,4): 320 (3,6): 160

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(40,2), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt6wtL2_order40_k40.txt, line 1. nonidentity supports a=[5], b=[5, 6, 15].

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

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

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 = 2.00.

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_k40.txt, line 1. This row is SmallGroup(40,2) with nonidentity GAP supports a=[5], b=[5, 6, 15]. 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=40), 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(40,2), 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=[5], b=[5, 6, 15] 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) = 40. 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, 4, 40, 44, 75, 79] [1, 7, 41, 47, 64, 73] [2, 10, 42, 46, 50, 55] [3, 11, 43, 51, 68, 76] [0, 4, 40, 44, 75, 79] [5, 14, 40, 44, 45, 54] [6, 15, 46, 55, 72, 78] [1, 7, 41, 47, 64, 73] [8, 18, 48, 53, 58, 63] [9, 19, 41, 47, 49, 59] [2, 10, 42, 46, 50, 55] [3, 11, 43, 51, 68, 76] [12, 22, 42, 50, 52, 62] [13, 23, 43, 51, 53, 63] [5, 14, 40, 44, 45, 54] [6, 15, 46, 55, 72, 78] [16, 26, 56, 61, 66, 71] [17, 27, 45, 54, 57, 67] [8, 18, 48, 53, 58, 63] [9, 19, 41, 47, 49, 59] [20, 30, 48, 58, 60, 70] [21, 31, 49, 59, 61, 71] [12, 22, 42, 50, 52, 62] [13, 23, 43, 51, 53, 63] [24, 33, 64, 69, 73, 77] [25, 34, 52, 62, 65, 74] [16, 26, 56, 61, 66, 71] [17, 27, 45, 54, 57, 67] [28, 36, 56, 66, 68, 76] [29, 37, 57, 67, 69, 77] [20, 30, 48, 58, 60, 70] [21, 31, 49, 59, 61, 71] [32, 38, 60, 70, 72, 78] [24, 33, 64, 69, 73, 77] [25, 34, 52, 62, 65, 74] [35, 39, 65, 74, 75, 79] [28, 36, 56, 66, 68, 76] [29, 37, 57, 67, 69, 77] [32, 38, 60, 70, 72, 78] [35, 39, 65, 74, 75, 79]

H_Z (40 checks, sparse supports)

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

Code ID 80-40-2 · download JSON · raw on GitHub