[[80,6,10]] · QEC Challenge

[[80,6,10]] d =

kd²/n

7.5

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 10, d_Z = 10 · 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 10 · witness weight 10 (claimed upper_bound)

witness operator (support, 10 qubits)

[15, 32, 45, 65, 67, 68, 69, 70, 71, 79]

d_Z 10 · witness weight 10 (claimed upper_bound)

witness operator (support, 10 qubits)

[5, 8, 15, 29, 34, 37, 45, 48, 53, 74]

certificate exact, d = 10 · CryptoMiniSat 5.14 SAT

X: no logical < 10 exists; Z: no logical < 10 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 6 · H_Z 6 (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 (2,2)×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 (1,4): 40 (2,2): 40 (2,4): 320 (2,6): 240 (3,2): 40 (3,4): 1360 (3,6): 3600 (3,8): 2160 (3,10): 160

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 (1,4): 40 (2,2): 40 (2,4): 320 (2,6): 240 (3,2): 40 (3,4): 1360 (3,6): 3600 (3,8): 2160 (3,10): 160

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction contributed via qldpc submit

model DeepSeek V4 Flash 0731 (claimed, not verified)

date 2026-09-18

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

[[80,6,10]] — two-block group-algebra code on SmallGroup(40,9)

Direction & hypothesis

Target: unrestricted weight-6 frontier for hackathon issue #1155. Reconstruct a published 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 7.50.

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_k6.txt, line 2. This row is SmallGroup(40,9) with nonidentity GAP supports a=[9], b=[2, 18, 30]. 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=6), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 10, not an exact certificate. The published distance (d=10) 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

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

H_Z (40 checks, sparse supports)

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

Code ID 80-6-10 · download JSON · raw on GitHub