[[96,50,2]] · QEC Challenge

[[96,50,2]] d =

kd²/n

2.083

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · 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 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[1, 8]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[16, 30]

certificate exact, d = 2 · scipy/HiGHS MILP

X: no logical < 2 exists; Z: no logical < 2 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 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 4 · H_Z 4

trapping sets H_X (1,4)×96 (2,0)×48 (3,0)×64 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 96 (2,0): 48 (2,4): 576 (3,0): 64 (3,4): 3840 (3,8): 768

trapping sets H_Z (1,4)×96 (2,0)×48 (3,0)×64 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 96 (2,0): 48 (2,4): 576 (3,0): 64 (3,4): 3840 (3,8): 768

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_wt8wtL4_order48_k50.txt, line 1. nonidentity supports a=[2, 5, 9], b=[5, 6, 16].

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

[[96,50,2]] — 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 = 2.08.

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_wt8wtL4_order48_k50.txt, line 1. This row is SmallGroup(48,1) with nonidentity GAP supports a=[2, 5, 9], b=[5, 6, 16]. 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=50), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 2, 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, 5, 9], b=[5, 6, 16] 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) = 50. 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, 4, 17, 31, 48, 52, 64, 78] [0, 1, 4, 8, 49, 56, 71, 85] [1, 2, 8, 11, 50, 59, 75, 88] [3, 6, 13, 18, 51, 61, 77, 89] [0, 4, 17, 31, 48, 52, 64, 78] [5, 15, 32, 42, 48, 52, 53, 63] [2, 6, 11, 18, 54, 66, 82, 92] [3, 7, 13, 20, 55, 68, 84, 93] [0, 1, 4, 8, 49, 56, 71, 85] [5, 9, 15, 22, 49, 56, 57, 70] [7, 10, 20, 24, 58, 72, 87, 94] [1, 2, 8, 11, 50, 59, 75, 88] [9, 12, 22, 26, 50, 59, 60, 74] [3, 6, 13, 18, 51, 61, 77, 89] [14, 19, 28, 33, 51, 61, 62, 76] [5, 15, 32, 42, 48, 52, 53, 63] [16, 30, 43, 47, 53, 63, 64, 78] [10, 17, 24, 31, 65, 79, 91, 95] [2, 6, 11, 18, 54, 66, 82, 92] [12, 19, 26, 33, 54, 66, 67, 81] [3, 7, 13, 20, 55, 68, 84, 93] [14, 21, 28, 35, 55, 68, 69, 83] [5, 9, 15, 22, 49, 56, 57, 70] [16, 23, 30, 37, 57, 70, 71, 85] [7, 10, 20, 24, 58, 72, 87, 94] [21, 25, 35, 38, 58, 72, 73, 86] [9, 12, 22, 26, 50, 59, 60, 74] [23, 27, 37, 40, 60, 74, 75, 88] [14, 19, 28, 33, 51, 61, 62, 76] [29, 34, 41, 44, 62, 76, 77, 89] [16, 30, 43, 47, 53, 63, 64, 78] [10, 17, 24, 31, 65, 79, 91, 95] [25, 32, 38, 42, 65, 79, 80, 90] [12, 19, 26, 33, 54, 66, 67, 81] [27, 34, 40, 44, 67, 81, 82, 92] [14, 21, 28, 35, 55, 68, 69, 83] [29, 36, 41, 45, 69, 83, 84, 93] [16, 23, 30, 37, 57, 70, 71, 85] [21, 25, 35, 38, 58, 72, 73, 86] [36, 39, 45, 46, 73, 86, 87, 94] [23, 27, 37, 40, 60, 74, 75, 88] [29, 34, 41, 44, 62, 76, 77, 89] [25, 32, 38, 42, 65, 79, 80, 90] [39, 43, 46, 47, 80, 90, 91, 95] [27, 34, 40, 44, 67, 81, 82, 92] [29, 36, 41, 45, 69, 83, 84, 93] [36, 39, 45, 46, 73, 86, 87, 94] [39, 43, 46, 47, 80, 90, 91, 95]

H_Z (48 checks, sparse supports)

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

Code ID 96-50-2 · download JSON · raw on GitHub