[[48,4,8]] · QEC Challenge

[[48,4,8]] d ≤

kd²/n

5.33333

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)

[9, 14, 17, 29, 33, 38, 40, 41]

d_Z 8 · witness weight 8 (claimed upper_bound)

witness operator (support, 8 qubits)

[0, 5, 7, 14, 24, 26, 33, 47]

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 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 3 · H_Z 3

trapping sets H_X (1,3)×48 (2,2)×24 (3,3)×288 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 48 (2,2): 24 (2,4): 312 (3,3): 288 (3,5): 2688 (3,7): 312

trapping sets H_Z (1,3)×48 (2,2)×24 (3,3)×288 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 48 (2,2): 24 (2,4): 312 (3,3): 288 (3,5): 2688 (3,7): 312

Construction & provenance

authors @mathysrennela

provenance submitted through the challenge

novelty novelty not audited

construction contributed via qldpc submit

model Beam 1.0 (claimed, not verified)

date 2026-11-09

notes WL-equivalent representative from a different space-group construction (arXiv:2606.20548); submitted as a distinct construction.

family other (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

[[48,4,8]] — Space-group topological code

Direction & hypothesis

Topological CSS code from a space group (arXiv:2606.20548). Targets the weight-6 unrestricted cell.

What was searched

Reconstructed via group-algebra permutation representations; CSS commutation and k verified. Distance matches the paper's SAT-certified values.

Evidence trail

Distance d=8 (upper bound). Gate passed (passed: true). Board-advancing per gate novelty.

Dead ends

Other rows in the paper's table did not commute under the standard construction.

Tools

Reconstructed with GAP/Python. Model: Beam 1.0.

Reproduction

arXiv:2606.20548, Table row [[48,4,8]].

Benchmark comparison (optional, Tier B/C)

Not applicable. EOF

Parity checks

X-checks 24 (max weight 6) · Z-checks 24 (max weight 6)

H_X (24 checks, sparse supports)

[1, 16, 22, 25, 32, 43] [2, 17, 21, 26, 33, 40] [3, 18, 20, 27, 34, 41] [0, 19, 23, 24, 35, 42] [12, 18, 21, 28, 39, 45] [13, 17, 22, 29, 36, 46] [14, 16, 23, 30, 37, 47] [15, 19, 20, 31, 38, 44] [8, 14, 17, 24, 35, 41] [9, 13, 18, 25, 32, 42] [10, 12, 19, 26, 33, 43] [11, 15, 16, 27, 34, 40] [4, 10, 13, 31, 37, 44] [5, 9, 14, 28, 38, 45] [6, 8, 15, 29, 39, 46] [7, 11, 12, 30, 36, 47] [0, 6, 9, 27, 33, 40] [1, 5, 10, 24, 34, 41] [2, 4, 11, 25, 35, 42] [3, 7, 8, 26, 32, 43] [2, 5, 20, 29, 36, 47] [1, 6, 21, 30, 37, 44] [0, 7, 22, 31, 38, 45] [3, 4, 23, 28, 39, 46]

H_Z (24 checks, sparse supports)

[3, 8, 17, 27, 40, 46] [0, 9, 18, 24, 41, 45] [1, 10, 19, 25, 42, 44] [2, 11, 16, 26, 43, 47] [4, 13, 23, 36, 42, 47] [5, 14, 20, 37, 41, 44] [6, 15, 21, 38, 40, 45] [7, 12, 22, 39, 43, 46] [0, 9, 19, 32, 38, 43] [1, 10, 16, 33, 37, 40] [2, 11, 17, 34, 36, 41] [3, 8, 18, 35, 39, 42] [5, 15, 20, 28, 34, 39] [6, 12, 21, 29, 33, 36] [7, 13, 22, 30, 32, 37] [4, 14, 23, 31, 35, 38] [1, 11, 16, 24, 30, 35] [2, 8, 17, 25, 29, 32] [3, 9, 18, 26, 28, 33] [0, 10, 19, 27, 31, 34] [7, 12, 21, 26, 31, 44] [4, 13, 22, 25, 28, 45] [5, 14, 23, 24, 29, 46] [6, 15, 20, 27, 30, 47]

Code ID 48-4-8-b · download JSON · raw on GitHub