[[96,22,6]] · QEC Challenge

[[96,22,6]] d =

kd²/n

8.25

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 6, d_Z = 6 · 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 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[10, 25, 39, 58, 73, 87]

d_Z 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[7, 10, 11, 18, 20, 31]

certificate exact, d = 6 · scipy/HiGHS MILP

X: no logical < 6 exists; Z: no logical < 6 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 (2,4)×64 (3,4)×32 (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 (2,4): 64 (2,6): 1216 (3,4): 32 (3,6): 3200 (3,8): 19264 (3,10): 1920

trapping sets H_Z (1,4)×96 (2,4)×64 (3,4)×32 (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 (2,4): 64 (2,6): 1216 (3,4): 32 (3,6): 3200 (3,8): 19264 (3,10): 1920

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(48,11), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL4_order48_k22.txt, line 20. nonidentity supports a=[2, 3, 32], b=[2, 13, 43].

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

[[96,22,6]] — two-block group-algebra code on SmallGroup(48,11)

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 = 8.25.

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_k22.txt, line 20. This row is SmallGroup(48,11) with nonidentity GAP supports a=[2, 3, 32], b=[2, 13, 43]. 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=22), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 6, 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(48,11), 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, 3, 32], b=[2, 13, 43] 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) = 22. 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, 6, 7, 11, 48, 55, 67, 88] [0, 1, 10, 18, 48, 49, 73, 92] [0, 2, 8, 17, 50, 64, 65, 70] [1, 3, 17, 24, 49, 51, 80, 94] [2, 4, 18, 20, 52, 68, 75, 81] [5, 19, 21, 26, 53, 54, 59, 84] [1, 2, 6, 13, 50, 54, 71, 76] [2, 3, 7, 31, 51, 55, 60, 95] [4, 6, 8, 24, 52, 56, 82, 86] [5, 9, 25, 33, 57, 58, 64, 66] [3, 6, 10, 20, 54, 58, 77, 83] [1, 4, 11, 31, 57, 59, 78, 79] [5, 12, 22, 32, 48, 56, 60, 91] [8, 10, 13, 31, 56, 61, 87, 90] [9, 14, 32, 38, 62, 65, 71, 72] [12, 15, 33, 35, 50, 63, 66, 93] [16, 34, 36, 40, 64, 69, 74, 82] [4, 7, 10, 17, 58, 63, 65, 84] [3, 8, 11, 18, 59, 62, 66, 85] [9, 12, 19, 28, 49, 61, 67, 75] [11, 13, 17, 20, 61, 68, 74, 91] [12, 14, 21, 42, 50, 69, 77, 79] [15, 19, 22, 38, 54, 70, 72, 78] [16, 23, 39, 44, 53, 71, 81, 87] [7, 13, 18, 24, 66, 69, 72, 89] [14, 19, 25, 35, 51, 68, 73, 82] [9, 15, 26, 42, 49, 52, 74, 95] [16, 27, 37, 43, 53, 75, 80, 85] [22, 25, 28, 42, 58, 76, 79, 85] [23, 29, 43, 46, 57, 77, 86, 91] [27, 30, 44, 45, 60, 78, 83, 92] [0, 20, 24, 31, 53, 72, 79, 93] [15, 21, 25, 32, 52, 55, 80, 87] [14, 22, 26, 33, 51, 56, 81, 88] [23, 27, 34, 41, 57, 60, 82, 89] [26, 28, 32, 35, 59, 65, 83, 89] [27, 29, 36, 47, 62, 75, 84, 90] [30, 34, 37, 46, 63, 67, 85, 94] [21, 28, 33, 38, 55, 61, 86, 92] [29, 34, 39, 45, 62, 67, 87, 93] [23, 30, 40, 47, 63, 71, 88, 90] [37, 39, 41, 47, 70, 73, 89, 95] [5, 35, 38, 42, 48, 68, 90, 94] [30, 36, 39, 43, 69, 73, 78, 91] [29, 37, 40, 44, 70, 74, 77, 92] [40, 41, 43, 45, 76, 80, 88, 93] [36, 41, 44, 46, 76, 81, 84, 94] [16, 45, 46, 47, 64, 83, 86, 95]

H_Z (48 checks, sparse supports)

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

Code ID 96-22-6 · download JSON · raw on GitHub