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[[56,6,7]] d =
n
56
k
6
d
7
kd²/n
5.25
w
6
X/Z
1
g
0.0026
r
6.7082
layers
2
swaps
291

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Distance

X/Z asymmetry 1 · d_X = 7, d_Z = 7 · 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 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[6, 14, 21, 28, 33, 42, 55]
d_Z 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[15, 18, 20, 21, 31, 42, 44]
certificate exact, d = 7 · scipy/HiGHS MILP
X: no logical < 7 exists; Z: no logical < 7 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)×28 (2,2)×28 (3,2)×28 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 28 (1,4): 28 (2,2): 28 (2,4): 224 (2,6): 168 (3,2): 28 (3,4): 1036 (3,6): 2324 (3,8): 1344 (3,10): 112
trapping sets H_Z (1,2)×28 (2,2)×28 (3,2)×28 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 28 (1,4): 28 (2,2): 28 (2,4): 224 (2,6): 168 (3,2): 28 (3,4): 1036 (3,6): 2324 (3,8): 1344 (3,10): 112
witness diameter X 6.7082 · Z 5.831 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
r = 6.708
X checkZ checkqubit site (28)2 qubits stacked (2 layers)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release
routing cost 291 nearest-neighbor SWAPs per round in total, at most 8 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

Construction & provenance

authors @vprusso and @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Two-block group-algebra (2BGA) code on the non-abelian group Dic7; n=56, k=6, max check weight 6.
model Claude Claude Opus 4.8 (claimed, not verified)
date 2026-08-04
notes Layout certification: a verifier-accepted honest bilayer layout (stacked 4x7 grid, 2 qubits per site, layers=2, measured interaction radius ~6.708) was found by @mathysrennela (Aug 2026). It earns the code the local-2d-bilayer class and makes the site's geometric efficiency g = 4kd^2/(n rho^2 r^4) computable. Co-authored by @mathysrennela for this layout contribution; code authorship shared with @vprusso.
family 2BGA coset (a tag, not a ranking)
locality 2D-local bilayer (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

[[56,6,7]] — 2BGA on the dicyclic Dic_7 group

Direction & hypothesis

Advance the frontier with a two-block group-algebra code over a non-abelian dicyclic (generalized-quaternion) group, a family that was not represented on the board when this search began. The distance was deliberately kept in the range where the RIS surrogate converges, so the claim is verifiable rather than an optimistic screen.

What was searched

2BGA H_X = [L(a) | R(b)], H_Z = [R(b)^T | L(a)^T] over a roster of dicyclic, dihedral and metacyclic groups, sampling element sets a, b under a check-weight bound and keeping only codes that strictly dominate an existing board entry in their (locality, weight) cell. This code sits on dicyclic Dic_7 with max check weight 6 and k = 6.

Evidence trail

Witness-backed upper bound d <= 7, with an explicit weight-7 logical on each side. The code was rebuilt from its group data and re-witnessed independently of the search that found it, and the distance held under a deeper independent RIS pass returning the same 7. verify/qldpc_verify.py accepts it at kd^2/n = 5.2. It strictly dominates 5 existing board entries.

Dead ends

The same search produced higher-distance candidates that did not survive scrutiny and are deliberately not submitted. A [[390,82,41]] screen resolved to d = 38 once measured at a budget calibrated against a known-answer control, matching the existing record rather than beating it, and several [[310,16,28-29]] screens fell to <= 25. Everything submitted here sits in the distance range where repeated independent passes agree.

Model & harness

Found by a continual non-abelian 2BGA dominance search (Claude Opus 4.8) built on this repository's own research/kit group-algebra constructors and the gf2_fast RIS core. Rebuilt, re-witnessed, and re-verified through verify/qldpc_verify.py during packaging.

Reproduction

Build dicyclic Dic_7 with research/kit/group_algebra (dicyclic presentation a^{2m} = 1, b^2 = a^m, b a b^-1 = a^-1), then form the 2BGA via build_2bga(mul, a, b) using the a, b element-index lists recorded in the code file to obtain [[56,6,7]].

---

Layout certification (co-authored)

Subsequent to submission, an honest bilayer layout was found and certified:

  • Layout: stacked 4x7 unit grid, layers = 2; qubits i and i+28 share a

    • site. Verified interaction radius r = 6.7082 (<= 7.0 bilayer cap), 2 qubits per site, min site spacing 1.0.

  • Effect: the trusted verifier now computes `locality_class =

    • local-2d-bilayer (the code was previously unrestricted` for lack of a layout), so it competes in the 2D-local cell and the site's geometric efficiency g = 4kd^2/(n rho^2 r^4) becomes defined: g ~ 0.0026 (upper-bound distance tier inherited), kd^2/n = 5.25.

  • Distance unchanged: d = 7 upper bound, witnesses intact (this edit only

    • adds locality and provenance — no checks or distance fields touched).

  • Authorship: @mathysrennela co-authored this layout contribution; the code

    • remains jointly credited with @vprusso. Layout found Aug 2026.

Reproduce the layout: `locality = {"coordinates": [[float((i % 28) % 4), float((i % 28) // 4)] for i in range(56)], "layers": 2}` and validate with uv run python verify/qldpc_verify.py codes/56-6-7.json (expect locality_class = local-2d-bilayer, interaction_radius = 6.7082).

Parity checks

X-checks 28 (max weight 6) · Z-checks 28 (max weight 6)
H_X (28 checks, sparse supports)
[17, 23, 24, 26, 39, 51] [2, 8, 25, 27, 32, 48] [0, 15, 21, 26, 41, 53] [0, 1, 6, 27, 34, 50] [0, 2, 13, 19, 43, 55] [1, 3, 4, 26, 36, 52] [2, 4, 11, 17, 29, 45] [2, 3, 5, 24, 38, 54] [4, 6, 9, 15, 31, 47] [0, 5, 7, 22, 28, 40] [6, 7, 8, 13, 33, 49] [7, 9, 20, 26, 30, 42] [5, 8, 10, 11, 35, 51] [9, 11, 18, 24, 32, 44] [3, 9, 10, 12, 37, 53] [11, 13, 16, 22, 34, 46] [1, 7, 12, 14, 39, 55] [13, 14, 15, 20, 36, 48] [5, 14, 16, 27, 29, 41] [12, 15, 17, 18, 38, 50] [3, 16, 18, 25, 31, 43] [10, 16, 17, 19, 40, 52] [1, 18, 20, 23, 33, 45] [8, 14, 19, 21, 42, 54] [20, 21, 22, 27, 35, 47] [6, 12, 21, 23, 28, 44] [19, 22, 24, 25, 37, 49] [4, 10, 23, 25, 30, 46]
H_Z (28 checks, sparse supports)
[9, 25, 30, 31, 32, 37] [6, 18, 31, 33, 44, 50] [11, 27, 29, 32, 34, 35] [8, 20, 33, 35, 42, 48] [1, 13, 33, 34, 36, 55] [10, 22, 35, 37, 40, 46] [3, 15, 31, 36, 38, 53] [12, 24, 37, 38, 39, 44] [5, 17, 29, 38, 40, 51] [14, 26, 36, 39, 41, 42] [7, 19, 40, 42, 49, 55] [0, 16, 34, 40, 41, 43] [9, 21, 42, 44, 47, 53] [2, 18, 32, 38, 43, 45] [11, 23, 44, 45, 46, 51] [4, 20, 30, 36, 45, 47] [13, 25, 43, 46, 48, 49] [6, 22, 28, 34, 47, 49] [15, 27, 41, 47, 48, 50] [8, 24, 32, 49, 51, 54] [1, 17, 39, 45, 50, 52] [10, 26, 30, 51, 52, 53] [3, 19, 37, 43, 52, 54] [0, 12, 28, 50, 53, 55] [5, 21, 28, 35, 41, 54] [2, 14, 29, 48, 54, 55] [7, 23, 28, 30, 33, 39] [4, 16, 29, 31, 46, 52]
Code ID 56-6-7 · download JSON · raw on GitHub