[[88,24,4]] · QEC Challenge

[[88,24,4]] d =

kd²/n

4.364

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 4, d_Z = 4 · 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 4 · witness weight 4 (claimed upper_bound)

witness operator (support, 4 qubits)

[27, 28, 67, 69]

d_Z 4 · witness weight 4 (claimed upper_bound)

witness operator (support, 4 qubits)

[15, 19, 21, 28]

certificate exact, d = 4 · scipy/HiGHS MILP

X: no logical < 4 exists; Z: no logical < 4 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 (2,4)×152 (3,2)×8 (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 (2,4): 152 (2,6): 928 (3,2): 8 (3,4): 688 (3,6): 2960 (3,8): 13728 (3,10): 912

trapping sets H_Z (1,4)×88 (2,4)×152 (3,2)×8 (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 (2,4): 152 (2,6): 928 (3,2): 8 (3,4): 688 (3,6): 2960 (3,8): 13728 (3,10): 912

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_k24.txt, line 2. nonidentity supports a=[2, 4, 9], b=[2, 8, 13].

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

[[88,24,4]] — 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 = 4.36.

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_k24.txt, line 2. This row is SmallGroup(44,1) with nonidentity GAP supports a=[2, 4, 9], b=[2, 8, 13]. 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=24), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 4, 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, 4, 9], b=[2, 8, 13] 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) = 24. 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, 4, 5, 39, 44, 48, 53, 79] [0, 1, 5, 42, 44, 45, 54, 81] [1, 2, 8, 42, 45, 46, 56, 82] [0, 3, 8, 9, 47, 49, 83, 87] [2, 4, 8, 39, 46, 48, 51, 84] [2, 3, 5, 9, 49, 50, 83, 85] [2, 5, 6, 12, 50, 52, 85, 86] [3, 7, 12, 13, 44, 45, 51, 84] [0, 6, 8, 12, 47, 52, 86, 87] [6, 7, 9, 13, 45, 46, 53, 79] [6, 9, 10, 16, 46, 48, 54, 81] [7, 11, 16, 17, 47, 55, 80, 85] [3, 10, 12, 16, 44, 48, 56, 82] [10, 11, 13, 17, 49, 57, 75, 86] [10, 13, 14, 20, 50, 58, 77, 87] [11, 15, 20, 21, 51, 59, 76, 81] [7, 14, 16, 20, 52, 60, 78, 83] [14, 15, 17, 21, 53, 61, 71, 82] [14, 17, 18, 24, 54, 62, 73, 84] [15, 19, 24, 25, 55, 63, 72, 77] [11, 18, 20, 24, 56, 64, 74, 79] [18, 19, 21, 25, 57, 65, 67, 78] [18, 21, 22, 28, 58, 66, 69, 80] [19, 23, 28, 29, 59, 67, 68, 73] [15, 22, 24, 28, 60, 68, 70, 75] [22, 23, 25, 29, 61, 63, 69, 74] [22, 25, 26, 32, 62, 65, 70, 76] [23, 27, 32, 33, 63, 64, 69, 71] [19, 26, 28, 32, 64, 66, 71, 72] [26, 27, 29, 33, 59, 65, 70, 73] [26, 29, 30, 36, 61, 66, 72, 74] [27, 31, 36, 37, 60, 65, 67, 75] [23, 30, 32, 36, 62, 67, 68, 76] [30, 31, 33, 37, 55, 66, 69, 77] [30, 33, 34, 40, 57, 68, 70, 78] [31, 35, 40, 41, 56, 61, 71, 79] [27, 34, 36, 40, 58, 63, 72, 80] [34, 35, 37, 41, 51, 62, 73, 81] [34, 37, 38, 43, 53, 64, 74, 82] [1, 35, 39, 43, 52, 57, 75, 83] [31, 38, 40, 43, 54, 59, 76, 84] [1, 38, 39, 41, 47, 58, 77, 85] [4, 38, 41, 42, 49, 60, 78, 86] [4, 35, 42, 43, 50, 55, 80, 87]

H_Z (44 checks, sparse supports)

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

Code ID 88-24-4 · download JSON · raw on GitHub