[[96,2,14]] · QEC Challenge

[[96,2,14]] d ≤

kd²/n

4.083

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 14, d_Z ≤ 14 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 14 · witness weight 14 (claimed upper_bound)

witness operator (support, 14 qubits)

[1, 7, 10, 19, 30, 44, 46, 49, 57, 74, 80, 86, 87, 93]

d_Z 14 · witness weight 14 (claimed upper_bound)

witness operator (support, 14 qubits)

[0, 10, 14, 18, 20, 21, 31, 33, 55, 66, 73, 82, 83, 91]

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 8 · H_Z 8

qubit degrees H_X 2–6 (mean 4.0) · H_Z 2–6 (mean 4.0)

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

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

(1,2): 48 (1,6): 48 (2,2): 48 (2,4): 32 (2,6): 512 (2,8): 96 (2,10): 528 (3,2): 48 (3,4): 272 (3,6): 2944 (3,8): 2448 (3,10): 10432 (3,12): 4096 (3,14): 5424 (3,16): 384

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

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

(1,2): 48 (1,6): 48 (2,2): 48 (2,4): 32 (2,6): 512 (2,8): 96 (2,10): 528 (3,2): 48 (3,4): 272 (3,6): 2944 (3,8): 2448 (3,10): 10432 (3,12): 4096 (3,14): 5424 (3,16): 384

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(48,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL2_order48_k2.txt, line 5. nonidentity supports a=[2], b=[2, 3, 11, 16, 31].

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

[[96,2,14]] — two-block group-algebra code on SmallGroup(48,1)

Direction & hypothesis

Target: unrestricted weight-8 frontier for hackathon issue #1155. Reconstruct a published weight-8 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 4.08.

What was searched

Scraped GitHub for downloadable parity-check matrix artifacts and parameter tables. From github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL2_order48_k2.txt, line 5. This row is SmallGroup(48,1) with nonidentity GAP supports a=[2], b=[2, 3, 11, 16, 31]. 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=2), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 14, not an exact certificate. The published distance 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(48,1), 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=[2], b=[2, 3, 11, 16, 31] 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) = 2. 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 8) · Z-checks 48 (max weight 8)

H_X (48 checks, sparse supports)

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

H_Z (48 checks, sparse supports)

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

Code ID 96-2-14 · download JSON · raw on GitHub