[[64,6,8]] · QEC Challenge

[[64,6,8]] d =

kd²/n

6.0

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 8, d_Z = 8 · 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 8 · witness weight 8 (claimed upper_bound)

witness operator (support, 8 qubits)

[29, 31, 33, 38, 39, 40, 48, 62]

d_Z 8 · witness weight 8 (claimed upper_bound)

witness operator (support, 8 qubits)

[14, 15, 23, 25, 43, 46, 52, 59]

certificate exact, d = 8 · scipy/HiGHS MILP

X: no logical < 8 exists; Z: no logical < 8 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 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)×32 (2,2)×32 (3,2)×32 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 32 (1,4): 32 (2,2): 32 (2,4): 256 (2,6): 192 (3,2): 32 (3,4): 1120 (3,6): 2848 (3,8): 1536 (3,10): 128

trapping sets H_Z (1,2)×32 (2,2)×32 (3,2)×32 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 32 (1,4): 32 (2,2): 32 (2,4): 256 (2,6): 192 (3,2): 32 (3,4): 1120 (3,6): 2848 (3,8): 1536 (3,10): 128

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

[[64,6,8]] — two-block group-algebra code on SmallGroup(32,1)

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 = 6.00.

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_order32_k6.txt, line 1. This row is SmallGroup(32,1) with nonidentity GAP supports a=[4], b=[2, 7, 24]. 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=6), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 8, not an exact certificate. The published distance (d=8) 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(32,1), 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=[4] and b=[2, 7, 24] 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 = 64 - rank(HX) - rank(HZ) = 6. 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 32 (max weight 6) · Z-checks 32 (max weight 6)

H_X (32 checks, sparse supports)

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

H_Z (32 checks, sparse supports)

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

Code ID 64-6-8 · download JSON · raw on GitHub