[[96,48,2]] · QEC Challenge

[[96,48,2]] d =

kd²/n

2.0

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · 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 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)

[14, 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 6 · H_Z 6

qubit degrees H_X 2–4 (mean 3.0) · H_Z 2–4 (mean 3.0)

trapping sets H_X (1,2)×48 (2,0)×48 (3,0)×192 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 48 (1,4): 48 (2,0): 48 (2,2): 192 (2,4): 96 (3,0): 192 (3,2): 480 (3,4): 384 (3,6): 192

trapping sets H_Z (1,2)×48 (2,0)×48 (3,0)×192 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 48 (1,4): 48 (2,0): 48 (2,2): 192 (2,4): 96 (3,0): 192 (3,2): 480 (3,4): 384 (3,6): 192

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_wt6wtL2_order48_k48.txt, line 1. nonidentity supports a=[6], b=[6, 7, 21].

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 ≤ 6 (computed)

How this code was found

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

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

Direction & hypothesis

Target: unrestricted weight-6 frontier for hackathon issue #1155. Reconstruct a published weight-6 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 2.00.

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_wt6wtL2_order48_k48.txt, line 1. This row is SmallGroup(48,2) with nonidentity GAP supports a=[6], b=[6, 7, 21]. 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=48), 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=[6], b=[6, 7, 21] 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) = 48. 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, 5, 48, 53, 90, 95] [1, 9, 49, 57, 58, 74] [2, 13, 50, 61, 69, 85] [3, 15, 51, 63, 65, 81] [4, 16, 52, 64, 79, 91] [0, 5, 48, 53, 90, 95] [6, 20, 48, 53, 54, 68] [7, 22, 55, 70, 72, 87] [8, 23, 56, 71, 73, 88] [1, 9, 49, 57, 58, 74] [10, 26, 58, 74, 82, 93] [11, 28, 49, 57, 59, 76] [12, 29, 55, 60, 70, 77] [2, 13, 50, 61, 69, 85] [14, 30, 62, 78, 80, 92] [3, 15, 51, 63, 65, 81] [4, 16, 52, 64, 79, 91] [17, 33, 50, 61, 65, 81] [18, 35, 51, 63, 66, 83] [19, 36, 52, 64, 67, 84] [6, 20, 48, 53, 54, 68] [21, 37, 69, 85, 86, 94] [7, 22, 55, 70, 72, 87] [8, 23, 56, 71, 73, 88] [24, 39, 54, 68, 72, 87] [25, 40, 66, 73, 83, 88] [10, 26, 58, 74, 82, 93] [27, 41, 56, 71, 75, 89] [11, 28, 49, 57, 59, 76] [12, 29, 55, 60, 70, 77] [14, 30, 62, 78, 80, 92] [31, 43, 59, 76, 79, 91] [32, 44, 60, 77, 80, 92] [17, 33, 50, 61, 65, 81] [34, 45, 62, 78, 82, 93] [18, 35, 51, 63, 66, 83] [19, 36, 52, 64, 67, 84] [21, 37, 69, 85, 86, 94] [38, 46, 67, 84, 86, 94] [24, 39, 54, 68, 72, 87] [25, 40, 66, 73, 83, 88] [27, 41, 56, 71, 75, 89] [42, 47, 75, 89, 90, 95] [31, 43, 59, 76, 79, 91] [32, 44, 60, 77, 80, 92] [34, 45, 62, 78, 82, 93] [38, 46, 67, 84, 86, 94] [42, 47, 75, 89, 90, 95]

H_Z (48 checks, sparse supports)

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

Code ID 96-48-2 · download JSON · raw on GitHub