[[78,4,11]] · QEC Challenge

[[78,4,11]] d ≤

kd²/n

6.205

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 11, d_Z ≤ 11 · w_X = 7, w_Z = 7 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 11 · witness weight 11 (claimed upper_bound)

witness operator (support, 11 qubits)

[5, 9, 16, 21, 23, 25, 29, 35, 66, 68, 75]

d_Z 11 · witness weight 11 (claimed upper_bound)

witness operator (support, 11 qubits)

[0, 3, 10, 20, 38, 45, 50, 58, 65, 69, 73]

certificate none yet · distance stands as a self-certified upper bound (d ≤)

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 7 · H_Z 7

qubit degrees H_X 2–5 (mean 3.5) · H_Z 2–5 (mean 3.5)

trapping sets H_X (1,2)×39 (2,2)×39 (3,2)×39 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 39 (1,5): 39 (2,2): 39 (2,5): 390 (2,6): 39 (2,8): 312 (3,2): 39 (3,5): 1950 (3,6): 780 (3,7): 429 (3,8): 5187 (3,9): 910 (3,10): 624 (3,11): 2418 (3,13): 195

trapping sets H_Z (1,2)×39 (2,2)×39 (3,2)×39 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 39 (1,5): 39 (2,2): 39 (2,5): 390 (2,6): 39 (2,8): 312 (3,2): 39 (3,5): 1950 (3,6): 780 (3,7): 429 (3,8): 5187 (3,9): 910 (3,10): 624 (3,11): 2418 (3,13): 195

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(39,2), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt7wtL2_order39_k4.txt, line 1. nonidentity supports a=[3], b=[2, 9, 10, 24].

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

[[78,4,11]] — two-block group-algebra code on SmallGroup(39,2)

Direction & hypothesis

Target: unrestricted weight-8 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 = 6.21.

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_wt7wtL2_order39_k4.txt, line 1. This row is SmallGroup(39,2) with nonidentity GAP supports a=[3], b=[2, 9, 10, 24]. 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=4), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 11, not an exact certificate. The published distance (d=11) 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(39,2), 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=[3], b=[2, 9, 10, 24] 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 = 78 - rank(HX) - rank(HZ) = 4. 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 39 (max weight 7) · Z-checks 39 (max weight 7)

H_X (39 checks, sparse supports)

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

H_Z (39 checks, sparse supports)

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

Code ID 78-4-11 · download JSON · raw on GitHub