[[76,4,10]] · QEC Challenge

[[76,4,10]] d =

kd²/n

5.263

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)

[1, 9, 20, 26, 37, 39, 44, 57, 60, 62]

d_Z 10 · witness weight 10 (claimed upper_bound)

witness operator (support, 10 qubits)

[4, 6, 11, 22, 29, 40, 46, 59, 67, 69]

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)×38 (2,2)×38 (3,2)×38 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 38 (1,4): 38 (2,2): 38 (2,4): 304 (2,6): 228 (3,2): 38 (3,4): 1330 (3,6): 3306 (3,8): 2052 (3,10): 152

trapping sets H_Z (1,2)×38 (2,2)×38 (3,2)×38 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 38 (1,4): 38 (2,2): 38 (2,4): 304 (2,6): 228 (3,2): 38 (3,4): 1330 (3,6): 3306 (3,8): 2052 (3,10): 152

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

[[76,4,10]] — two-block group-algebra code on SmallGroup(38,2)

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 = 5.26.

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_order38_k4.txt, line 1. This row is SmallGroup(38,2) with nonidentity GAP supports a=[3], b=[4, 14, 23]. 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 ≤ 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(38,2), 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] and b=[4, 14, 23] 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 = 76 - 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 38 (max weight 6) · Z-checks 38 (max weight 6)

H_X (38 checks, sparse supports)

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

H_Z (38 checks, sparse supports)

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

Code ID 76-4-10 · download JSON · raw on GitHub