[[96,12,9]] · QEC Challenge

[[96,12,9]] d =

kd²/n

10.125

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 9, d_Z = 9 · 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 9 · witness weight 9 (claimed upper_bound)

witness operator (support, 9 qubits)

[3, 9, 24, 51, 56, 82, 85, 87, 94]

d_Z 9 · witness weight 9 (claimed upper_bound)

witness operator (support, 9 qubits)

[6, 9, 15, 24, 30, 43, 51, 72, 82]

certificate exact, d = 9 · CryptoMiniSat 5.14 SAT

X: no logical < 9 exists; Z: no logical < 9 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 7 · H_Z 7

qubit degrees H_X 3–4 (mean 3.5) · H_Z 3–4 (mean 3.5)

trapping sets H_X (1,3)×48 (2,4)×144 (3,3)×48 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 48 (1,4): 48 (2,4): 144 (2,5): 576 (2,6): 288 (3,3): 48 (3,4): 144 (3,5): 624 (3,6): 4800 (3,7): 6960 (3,8): 2160 (3,9): 864 (3,10): 192

trapping sets H_Z (1,3)×48 (2,4)×144 (3,3)×48 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 48 (1,4): 48 (2,4): 144 (2,5): 576 (2,6): 288 (3,3): 48 (3,4): 144 (3,5): 624 (3,6): 4800 (3,7): 6960 (3,8): 2160 (3,9): 864 (3,10): 192

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 ≤ 8 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[96,12,9]] — two-block group-algebra code on SmallGroup(48,20)

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 = 10.12.

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_wt7wtL3_order48_k12.txt, line 2. This row is SmallGroup(48,20) with nonidentity GAP supports a=[8, 25], b=[2, 18, 38]. 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=12), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 9, not an exact certificate. The published distance (d=9) 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(48,20), 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=[8, 25] and b=[2, 18, 38] 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 = 96 - rank(HX) - rank(HZ) = 12. 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 48 (max weight 7) · Z-checks 48 (max weight 7)

H_X (48 checks, sparse supports)

[0, 26, 35, 48, 56, 84, 95] [1, 13, 33, 48, 49, 87, 89] [2, 3, 42, 50, 66, 83, 91] [3, 8, 39, 51, 57, 69, 82] [4, 20, 40, 49, 52, 91, 93] [5, 10, 45, 53, 68, 71, 90] [6, 7, 24, 50, 54, 61, 94] [0, 7, 43, 51, 55, 60, 64] [8, 28, 44, 52, 56, 77, 94] [9, 17, 29, 53, 57, 72, 76] [10, 13, 18, 56, 58, 67, 80] [11, 14, 31, 54, 59, 68, 95] [12, 15, 47, 60, 79, 82, 93] [12, 13, 21, 61, 70, 83, 92] [1, 14, 46, 55, 62, 67, 71] [15, 23, 24, 49, 63, 66, 85] [16, 25, 36, 57, 64, 79, 83] [2, 17, 20, 48, 58, 65, 75] [18, 21, 38, 59, 66, 76, 87] [19, 22, 39, 60, 67, 77, 86] [3, 19, 20, 61, 68, 74, 78] [4, 21, 47, 53, 62, 69, 75] [5, 22, 31, 50, 52, 63, 70] [23, 32, 41, 61, 64, 71, 86] [0, 24, 32, 69, 72, 81, 90] [6, 25, 28, 49, 65, 73, 82] [26, 29, 34, 54, 71, 74, 92] [27, 30, 43, 67, 75, 84, 90] [7, 27, 28, 68, 76, 81, 85] [2, 29, 37, 55, 77, 80, 93] [9, 30, 38, 54, 56, 70, 78] [1, 10, 31, 51, 72, 79, 88] [11, 32, 35, 52, 60, 73, 80] [12, 33, 36, 53, 59, 74, 81] [34, 37, 46, 72, 75, 82, 89] [14, 34, 35, 63, 76, 83, 88] [6, 15, 36, 58, 62, 77, 84] [16, 37, 42, 48, 59, 78, 85] [4, 17, 38, 55, 79, 86, 92] [5, 39, 44, 65, 85, 87, 95] [19, 40, 41, 57, 66, 81, 88] [11, 22, 41, 65, 69, 84, 89] [8, 25, 42, 62, 74, 86, 90] [9, 26, 43, 63, 73, 87, 91] [27, 44, 45, 50, 64, 88, 92] [18, 30, 45, 51, 73, 89, 93] [16, 33, 46, 70, 80, 91, 94] [23, 40, 47, 58, 78, 94, 95]

H_Z (48 checks, sparse supports)

[0, 1, 17, 37, 48, 55, 72] [1, 4, 15, 25, 49, 62, 79] [2, 6, 22, 44, 50, 65, 77] [3, 7, 31, 45, 50, 51, 68] [4, 8, 22, 32, 52, 69, 86] [5, 9, 21, 33, 53, 70, 87] [6, 11, 26, 30, 54, 73, 84] [7, 14, 29, 38, 54, 55, 76] [0, 8, 10, 30, 51, 56, 90] [3, 9, 16, 40, 57, 78, 91] [10, 17, 36, 47, 53, 58, 79] [11, 18, 33, 37, 59, 80, 89] [7, 12, 19, 32, 60, 61, 81] [6, 13, 20, 23, 49, 58, 61] [14, 21, 36, 42, 59, 62, 83] [15, 22, 35, 43, 60, 63, 84] [7, 16, 23, 44, 64, 85, 94] [17, 25, 39, 41, 57, 65, 86] [2, 15, 18, 40, 58, 66, 93] [10, 14, 19, 27, 67, 68, 88] [5, 11, 20, 28, 52, 65, 68] [3, 21, 24, 41, 61, 66, 69] [13, 22, 30, 46, 67, 70, 89] [5, 14, 23, 26, 63, 71, 95] [9, 24, 31, 34, 54, 63, 72] [25, 32, 43, 45, 64, 73, 90] [20, 26, 33, 42, 48, 74, 91] [17, 21, 27, 34, 75, 76, 92] [9, 18, 28, 35, 56, 73, 76] [8, 19, 29, 36, 57, 74, 77] [20, 30, 37, 47, 75, 78, 93] [12, 16, 31, 38, 59, 70, 79] [10, 29, 32, 46, 71, 72, 80] [24, 28, 33, 40, 49, 81, 94] [3, 12, 25, 34, 74, 82, 83] [2, 13, 16, 35, 48, 80, 83] [0, 27, 36, 41, 64, 81, 84] [15, 28, 37, 39, 77, 82, 85] [19, 23, 38, 42, 66, 78, 86] [1, 18, 39, 43, 51, 67, 87] [31, 35, 40, 44, 52, 88, 95] [1, 34, 41, 45, 71, 88, 89] [5, 24, 27, 42, 50, 85, 90] [2, 4, 43, 46, 55, 75, 91] [13, 26, 38, 44, 56, 87, 92] [4, 12, 29, 45, 53, 92, 93] [6, 8, 46, 47, 62, 82, 94] [0, 11, 39, 47, 60, 69, 95]

Code ID 96-12-9 · download JSON · raw on GitHub