[[88,46,2]] · QEC Challenge

[[88,46,2]] d =

kd²/n

2.091

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · 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 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[73, 76]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[25, 26]

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 8 · H_Z 8

qubit degrees H_X 4 · H_Z 4

trapping sets H_X (1,4)×88 (2,0)×88 (3,4)×3124 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 88 (2,0): 88 (2,4): 440 (3,4): 3124 (3,8): 352

trapping sets H_Z (1,4)×88 (2,0)×88 (3,4)×3124 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 88 (2,0): 88 (2,4): 440 (3,4): 3124 (3,8): 352

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(44,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL4_order44_k46.txt, line 1. nonidentity supports a=[2, 3, 5], b=[3, 4, 7].

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

[[88,46,2]] — two-block group-algebra code on SmallGroup(44,1)

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 = 2.09.

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_order44_k46.txt, line 1. This row is SmallGroup(44,1) with nonidentity GAP supports a=[2, 3, 5], b=[3, 4, 7]. 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=46), 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 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(44,1), 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, 5], b=[3, 4, 7] 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 = 88 - rank(HX) - rank(HZ) = 46. 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 44 (max weight 8) · Z-checks 44 (max weight 8)

H_X (44 checks, sparse supports)

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

H_Z (44 checks, sparse supports)

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

Code ID 88-46-2 · download JSON · raw on GitHub