[[68,36,2]] · QEC Challenge

[[68,36,2]] d =

kd²/n

2.118

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)

[9, 42]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[60, 61]

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)×68 (2,0)×102 (3,4)×1972 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 68 (2,0): 102 (2,4): 272 (3,4): 1972

trapping sets H_Z (1,4)×68 (2,0)×102 (3,4)×1972 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 68 (2,0): 102 (2,4): 272 (3,4): 1972

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(34,2), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt8wtL4_order34_k36.txt, line 1. nonidentity supports a=[2, 3, 4], b=[2, 3, 4].

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

[[68,36,2]] — two-block group-algebra code on SmallGroup(34,2)

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 = 2.12.

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_wt8wtL4_order34_k36.txt, line 1. This row is SmallGroup(34,2) with nonidentity GAP supports a=[2, 3, 4], b=[2, 3, 4]. 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=36), 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(34,2), 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=[2, 3, 4], b=[2, 3, 4] 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 = 68 - rank(HX) - rank(HZ) = 36. 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 34 (max weight 8) · Z-checks 34 (max weight 8)

H_X (34 checks, sparse supports)

[0, 1, 32, 33, 34, 35, 66, 67] [0, 1, 32, 33, 34, 35, 66, 67] [0, 1, 2, 3, 34, 35, 36, 37] [0, 1, 2, 3, 34, 35, 36, 37] [2, 3, 4, 5, 36, 37, 38, 39] [2, 3, 4, 5, 36, 37, 38, 39] [4, 5, 6, 7, 38, 39, 40, 41] [4, 5, 6, 7, 38, 39, 40, 41] [6, 7, 8, 9, 40, 41, 42, 43] [6, 7, 8, 9, 40, 41, 42, 43] [8, 9, 10, 11, 42, 43, 44, 45] [8, 9, 10, 11, 42, 43, 44, 45] [10, 11, 12, 13, 44, 45, 46, 47] [10, 11, 12, 13, 44, 45, 46, 47] [12, 13, 14, 15, 46, 47, 48, 49] [12, 13, 14, 15, 46, 47, 48, 49] [14, 15, 16, 17, 48, 49, 50, 51] [14, 15, 16, 17, 48, 49, 50, 51] [16, 17, 18, 19, 50, 51, 52, 53] [16, 17, 18, 19, 50, 51, 52, 53] [18, 19, 20, 21, 52, 53, 54, 55] [18, 19, 20, 21, 52, 53, 54, 55] [20, 21, 22, 23, 54, 55, 56, 57] [20, 21, 22, 23, 54, 55, 56, 57] [22, 23, 24, 25, 56, 57, 58, 59] [22, 23, 24, 25, 56, 57, 58, 59] [24, 25, 26, 27, 58, 59, 60, 61] [24, 25, 26, 27, 58, 59, 60, 61] [26, 27, 28, 29, 60, 61, 62, 63] [26, 27, 28, 29, 60, 61, 62, 63] [28, 29, 30, 31, 62, 63, 64, 65] [28, 29, 30, 31, 62, 63, 64, 65] [30, 31, 32, 33, 64, 65, 66, 67] [30, 31, 32, 33, 64, 65, 66, 67]

H_Z (34 checks, sparse supports)

[0, 1, 2, 3, 34, 35, 36, 37] [0, 1, 2, 3, 34, 35, 36, 37] [2, 3, 4, 5, 36, 37, 38, 39] [2, 3, 4, 5, 36, 37, 38, 39] [4, 5, 6, 7, 38, 39, 40, 41] [4, 5, 6, 7, 38, 39, 40, 41] [6, 7, 8, 9, 40, 41, 42, 43] [6, 7, 8, 9, 40, 41, 42, 43] [8, 9, 10, 11, 42, 43, 44, 45] [8, 9, 10, 11, 42, 43, 44, 45] [10, 11, 12, 13, 44, 45, 46, 47] [10, 11, 12, 13, 44, 45, 46, 47] [12, 13, 14, 15, 46, 47, 48, 49] [12, 13, 14, 15, 46, 47, 48, 49] [14, 15, 16, 17, 48, 49, 50, 51] [14, 15, 16, 17, 48, 49, 50, 51] [16, 17, 18, 19, 50, 51, 52, 53] [16, 17, 18, 19, 50, 51, 52, 53] [18, 19, 20, 21, 52, 53, 54, 55] [18, 19, 20, 21, 52, 53, 54, 55] [20, 21, 22, 23, 54, 55, 56, 57] [20, 21, 22, 23, 54, 55, 56, 57] [22, 23, 24, 25, 56, 57, 58, 59] [22, 23, 24, 25, 56, 57, 58, 59] [24, 25, 26, 27, 58, 59, 60, 61] [24, 25, 26, 27, 58, 59, 60, 61] [26, 27, 28, 29, 60, 61, 62, 63] [26, 27, 28, 29, 60, 61, 62, 63] [28, 29, 30, 31, 62, 63, 64, 65] [28, 29, 30, 31, 62, 63, 64, 65] [30, 31, 32, 33, 64, 65, 66, 67] [30, 31, 32, 33, 64, 65, 66, 67] [0, 1, 32, 33, 34, 35, 66, 67] [0, 1, 32, 33, 34, 35, 66, 67]

Code ID 68-36-2 · download JSON · raw on GitHub