[[96,16,6]] · QEC Challenge

[[96,16,6]] d =

kd²/n

6.0

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 6, d_Z = 6 · w_X = 6, w_Z = 6 (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)

[56, 59, 74, 81, 85, 92]

d_Z 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[51, 61, 63, 76, 77, 78]

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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 6 · H_Z 6

qubit degrees H_X 3 · H_Z 3

trapping sets H_X (1,3)×96 (2,4)×720 (3,3)×96 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 96 (2,4): 720 (3,3): 96 (3,5): 6912 (3,7): 960

trapping sets H_Z (1,3)×96 (2,4)×720 (3,3)×96 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 96 (2,4): 720 (3,3): 96 (3,5): 6912 (3,7): 960

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 ≤ 6 (computed)

How this code was found

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

[[96,16,6]] — two-block group-algebra code on SmallGroup(48,52)

Direction & hypothesis

Target: unrestricted weight-6 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 = 6.00.

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_wt6wtL3_order48_k16.txt, line 2. This row is SmallGroup(48,52) with nonidentity GAP supports a=[10, 28], b=[15, 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=16), 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 (d=6) 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(48,52), 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=[10, 28] and b=[15, 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) = 16. 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 6) · Z-checks 48 (max weight 6)

H_X (48 checks, sparse supports)

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

H_Z (48 checks, sparse supports)

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

Code ID 96-16-6 · download JSON · raw on GitHub