[[88,4,12]] · QEC Challenge

[[88,4,12]] d ≤

kd²/n

6.545

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 12, d_Z ≤ 12 · 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 12 · witness weight 12 (claimed upper_bound)

witness operator (support, 12 qubits)

[16, 20, 21, 35, 43, 46, 48, 52, 59, 67, 78, 81]

d_Z 12 · witness weight 12 (claimed upper_bound)

witness operator (support, 12 qubits)

[12, 14, 16, 20, 26, 31, 32, 48, 63, 66, 81, 87]

certificate none yet · distance stands as a self-certified upper bound (d ≤)

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 2–4 (mean 3.0) · H_Z 2–4 (mean 3.0)

trapping sets H_X (1,2)×44 (2,2)×44 (3,2)×44 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 44 (1,4): 44 (2,2): 44 (2,4): 352 (2,6): 264 (3,2): 44 (3,4): 1496 (3,6): 3960 (3,8): 2376 (3,10): 176

trapping sets H_Z (1,2)×44 (2,2)×44 (3,2)×44 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 44 (1,4): 44 (2,2): 44 (2,4): 352 (2,6): 264 (3,2): 44 (3,4): 1496 (3,6): 3960 (3,8): 2376 (3,10): 176

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

[[88,4,12]] — two-block group-algebra code on SmallGroup(44,4)

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 = 6.55.

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_order44_k4.txt, line 2. This row is SmallGroup(44,4) with nonidentity GAP supports a=[6], b=[7, 12, 29]. 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=4), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 12, not an exact certificate. The published distance (d=12) 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(44,4), 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] and b=[7, 12, 29] 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) = 4. 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 6) · Z-checks 44 (max weight 6)

H_X (44 checks, sparse supports)

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

H_Z (44 checks, sparse supports)

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

Code ID 88-4-12 · download JSON · raw on GitHub