[[88,44,2]] · QEC Challenge

[[88,44,2]] 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)

[76, 81]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[30, 35]

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)×44 (2,0)×44 (3,0)×176 (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 (2,0): 44 (2,2): 176 (2,4): 88 (3,0): 176 (3,2): 440 (3,4): 352 (3,6): 176

trapping sets H_Z (1,2)×44 (2,0)×44 (3,0)×176 (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 (2,0): 44 (2,2): 176 (2,4): 88 (3,0): 176 (3,2): 440 (3,4): 352 (3,6): 176

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(44,2), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt6wtL2_order44_k44.txt, line 1. nonidentity supports a=[4], b=[4, 5, 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 ≤ 6 (computed)

How this code was found

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

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

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_order44_k44.txt, line 1. This row is SmallGroup(44,2) with nonidentity GAP supports a=[4], b=[4, 5, 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=44), 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(44,2), 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=[4], b=[4, 5, 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 = 88 - rank(HX) - rank(HZ) = 44. 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, 3, 44, 47, 84, 87] [1, 5, 45, 49, 82, 86] [2, 7, 45, 46, 49, 51] [0, 3, 44, 47, 84, 87] [4, 9, 44, 47, 48, 53] [1, 5, 45, 49, 82, 86] [6, 11, 48, 50, 53, 55] [2, 7, 45, 46, 49, 51] [8, 13, 46, 51, 52, 57] [4, 9, 44, 47, 48, 53] [10, 15, 52, 54, 57, 59] [6, 11, 48, 50, 53, 55] [12, 17, 50, 55, 56, 61] [8, 13, 46, 51, 52, 57] [14, 19, 56, 58, 61, 63] [10, 15, 52, 54, 57, 59] [16, 21, 54, 59, 60, 65] [12, 17, 50, 55, 56, 61] [18, 23, 60, 62, 65, 67] [14, 19, 56, 58, 61, 63] [20, 25, 58, 63, 64, 69] [16, 21, 54, 59, 60, 65] [22, 27, 64, 66, 69, 71] [18, 23, 60, 62, 65, 67] [24, 29, 62, 67, 68, 73] [20, 25, 58, 63, 64, 69] [26, 31, 68, 70, 73, 75] [22, 27, 64, 66, 69, 71] [28, 33, 66, 71, 72, 77] [24, 29, 62, 67, 68, 73] [30, 35, 72, 74, 77, 79] [26, 31, 68, 70, 73, 75] [32, 37, 70, 75, 76, 81] [28, 33, 66, 71, 72, 77] [34, 39, 76, 78, 81, 83] [30, 35, 72, 74, 77, 79] [36, 41, 74, 79, 80, 85] [32, 37, 70, 75, 76, 81] [38, 42, 80, 82, 85, 86] [34, 39, 76, 78, 81, 83] [40, 43, 78, 83, 84, 87] [36, 41, 74, 79, 80, 85] [38, 42, 80, 82, 85, 86] [40, 43, 78, 83, 84, 87]

H_Z (44 checks, sparse supports)

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

Code ID 88-44-2 · download JSON · raw on GitHub