[[96,36,2]] · QEC Challenge

[[96,36,2]] d =

kd²/n

1.5

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · w_X = 7, w_Z = 7 (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)

[31, 43]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[1, 17]

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 7 · H_Z 7

qubit degrees H_X 3–4 (mean 3.5) · H_Z 3–4 (mean 3.5)

trapping sets H_X (1,3)×48 (2,0)×72 (3,3)×304 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 48 (1,4): 48 (2,0): 72 (2,4): 96 (2,5): 576 (3,3): 304 (3,4): 864 (3,5): 1152 (3,6): 2592 (3,7): 2880

trapping sets H_Z (1,3)×48 (2,0)×72 (3,3)×304 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 48 (1,4): 48 (2,0): 72 (2,4): 96 (2,5): 576 (3,3): 304 (3,4): 864 (3,5): 1152 (3,6): 2592 (3,7): 2880

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(48,2), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt7wtL3_order48_k36.txt, line 1. nonidentity supports a=[3, 11], b=[2, 6, 10].

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

[[96,36,2]] — two-block group-algebra code on SmallGroup(48,2)

Direction & hypothesis

Target: unrestricted weight-8 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 = 1.50.

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_wt7wtL3_order48_k36.txt, line 1. This row is SmallGroup(48,2) with nonidentity GAP supports a=[3, 11], b=[2, 6, 10]. 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=36), 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 (d=2) 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(48,2), 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=[3, 11], b=[2, 6, 10] 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) = 36. 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 7) · Z-checks 48 (max weight 7)

H_X (48 checks, sparse supports)

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

H_Z (48 checks, sparse supports)

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

Code ID 96-36-2 · download JSON · raw on GitHub