[[96,21,6]] · QEC Challenge

[[96,21,6]] d =

kd²/n

7.875

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)

[1, 8, 10, 21, 24, 46]

d_Z 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[65, 66, 82, 90, 92, 95]

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)×96 (3,4)×48 (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): 96 (2,6): 1152 (3,4): 48 (3,6): 4000 (3,8): 17088 (3,10): 1584

trapping sets H_Z (1,4)×96 (2,4)×96 (3,4)×192 (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): 96 (2,6): 1152 (3,4): 192 (3,6): 3520 (3,8): 17568 (3,10): 1536

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(48,3), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL4_order48_k21.txt, line 3. nonidentity supports a=[2, 3, 23], b=[3, 4, 29].

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

[[96,21,6]] — two-block group-algebra code on SmallGroup(48,3)

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 = 7.88.

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_k21.txt, line 3. This row is SmallGroup(48,3) with nonidentity GAP supports a=[2, 3, 23], b=[3, 4, 29]. 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=21), 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(48,3), 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, 23], b=[3, 4, 29] 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) = 21. 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, 29, 47, 48, 59, 62, 77] [0, 1, 13, 24, 49, 69, 72, 87] [0, 2, 17, 18, 48, 50, 75, 78] [3, 18, 33, 41, 48, 51, 60, 89] [4, 13, 19, 43, 51, 52, 61, 75] [5, 12, 20, 42, 53, 60, 76, 78] [1, 6, 31, 40, 54, 79, 82, 92] [2, 4, 7, 37, 49, 55, 85, 88] [1, 3, 8, 28, 49, 56, 70, 94] [4, 8, 9, 29, 56, 57, 71, 85] [2, 5, 10, 40, 58, 70, 86, 88] [3, 11, 19, 31, 50, 51, 53, 59] [4, 12, 32, 34, 52, 59, 60, 63] [5, 13, 33, 35, 53, 61, 62, 89] [14, 28, 34, 44, 50, 52, 62, 76] [15, 17, 27, 35, 53, 63, 75, 77] [2, 16, 31, 36, 50, 63, 64, 89] [7, 17, 45, 46, 54, 65, 90, 93] [8, 10, 18, 32, 54, 66, 80, 95] [9, 19, 25, 42, 66, 67, 81, 90] [10, 20, 24, 43, 68, 80, 91, 93] [7, 11, 14, 21, 55, 56, 58, 69] [0, 12, 21, 22, 57, 69, 70, 73] [13, 16, 23, 46, 58, 71, 72, 94] [9, 14, 24, 41, 55, 57, 72, 86] [10, 11, 15, 25, 58, 73, 85, 87] [12, 16, 25, 26, 55, 73, 74, 94] [6, 14, 27, 42, 60, 62, 64, 75] [15, 28, 36, 43, 48, 61, 63, 76] [16, 29, 44, 45, 51, 64, 76, 77] [11, 30, 32, 45, 59, 61, 64, 78] [20, 21, 23, 31, 65, 66, 68, 79] [22, 32, 35, 38, 67, 79, 80, 83] [23, 33, 34, 37, 68, 81, 82, 95] [17, 24, 26, 34, 65, 67, 82, 91] [1, 25, 35, 44, 68, 83, 90, 92] [8, 26, 36, 47, 65, 83, 84, 95] [3, 22, 27, 37, 70, 72, 74, 85] [23, 28, 30, 38, 49, 71, 73, 86] [5, 29, 38, 39, 56, 74, 86, 87] [26, 27, 30, 40, 69, 71, 74, 88] [20, 30, 41, 47, 52, 77, 78, 89] [36, 37, 39, 42, 80, 82, 84, 90] [6, 7, 38, 43, 54, 81, 83, 91] [18, 21, 39, 44, 66, 84, 91, 92] [9, 33, 40, 45, 79, 81, 84, 93] [15, 39, 41, 46, 57, 87, 88, 94] [19, 22, 46, 47, 67, 92, 93, 95]

H_Z (48 checks, sparse supports)

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

Code ID 96-21-6 · download JSON · raw on GitHub