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[[64,16,6]] d =
n
64
k
16
d
6
kd²/n
9.0
w
10
X/Z
1
g
0.0037
r
7.0
layers
2
swaps
293

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Distance

X/Z asymmetry 1 · d_X = 6, d_Z = 6 · w_X = 10, w_Z = 10 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[14, 22, 34, 59, 60, 61]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[8, 9, 11, 13, 50, 53]
certificate exact, d = 6 · CryptoMiniSat 5.14 SAT
X: no logical < 6 exists (CryptoMiniSat, XOR + sequential-counter cardinality); Z: no logical < 6 exists (CryptoMiniSat, XOR + sequential-counter cardinality)

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 10 · H_Z 10
qubit degrees H_X 4–6 (mean 5.0) · H_Z 4–6 (mean 5.0)
trapping sets H_X (1,4)×32 (2,4)×16 (3,4)×56 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 32 (1,6): 32 (2,4): 16 (2,6): 368 (2,8): 576 (2,10): 96 (3,4): 56 (3,6): 1072 (3,8): 6072 (3,10): 9400 (3,12): 4232 (3,14): 448
trapping sets H_Z (1,4)×32 (2,4)×16 (3,4)×56 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 32 (1,6): 32 (2,4): 16 (2,6): 368 (2,8): 576 (2,10): 96 (3,4): 56 (3,6): 1072 (3,8): 6072 (3,10): 9400 (3,12): 4232 (3,14): 448
witness diameter X 6.0828 · Z 6.0828 (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
layout contributed by @dorakingx · Simulated annealing over site assignments on an integer grid with the repo's own research/local2d/fold_layout.search_layout (max-diameter objective, several bounding boxes and seeds); single layer at radius 4 tried first, then two layers at 7. Radius and per-site occupancy re-measured with the verifier's rule. · 2026-10-02
r = 7
X checkZ checkqubit site (32)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 293 nearest-neighbor SWAPs per round in total, at most 10 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
provenance submitted through the challenge
novelty novelty not audited
construction Two-block group-algebra (2BGA) code on the non-abelian group meta(16,2,9); n=64, k=16, max check weight 10.
model Claude Claude Opus 4.8 (claimed, not verified)
date 2026-08-04
family 2BGA coset (a tag, not a ranking)
locality 2D-local bilayer (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

[[64,16,6]] — 2BGA on the metacyclic Z_16 x| Z_2 group

Direction & hypothesis

Advance the low-blocklength frontier with a two-block group-algebra code over a non-abelian group. Small n and modest distance were chosen deliberately: this is the regime where the RIS distance surrogate converges, so the claim is verifiable rather than an optimistic upper bound.

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 a, b with bounded check weight and keeping only codes that strictly dominate an existing board entry in their (locality, weight) cell. This code sits on the metacyclic Z_16 x| Z_2 group with max check weight 10.

Evidence trail

Witness-backed upper bound d <= 6, with an explicit weight-6 logical on each side. The code was rebuilt from its group data and re-witnessed independently of the search, and a deeper RIS pass returned the same distance rather than a lower one. verify/qldpc_verify.py accepts it at kd^2/n = 9.0. It strictly dominates 1 existing board entries.

Dead ends

The same search at high distance is not trustworthy: candidates screening near d ~ 40 collapsed under deeper search (one [[390,82,41]] screen resolved to 38 at 20M reads, matching the incumbent rather than beating it). Everything submitted here is confined to the low-distance regime where repeated deeper passes agree.

Model & harness

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

Reproduction

Build the group with research/kit/group_algebra (dicyclic presentation a^{2m}=1, b^2=a^m, bab^-1=a^-1 for Dic_m; metacyclic(p,k,r) otherwise), then build the 2BGA via build_2bga(mul, a, b) with the a, b element-index lists recorded in the code file to obtain [[64,16,6]].

Parity checks

X-checks 32 (max weight 10) · Z-checks 32 (max weight 10)
H_X (32 checks, sparse supports)
[7, 9, 17, 30, 50, 52, 54, 55, 58, 60] [6, 8, 16, 31, 35, 38, 39, 43, 53, 61] [0, 3, 25, 27, 52, 54, 56, 57, 60, 62] [1, 2, 24, 26, 37, 40, 41, 45, 55, 63] [2, 11, 13, 21, 32, 54, 56, 58, 59, 62] [3, 10, 12, 20, 33, 39, 42, 43, 47, 57] [4, 7, 29, 31, 32, 34, 56, 58, 60, 61] [5, 6, 28, 30, 35, 41, 44, 45, 49, 59] [6, 15, 17, 25, 34, 36, 58, 60, 62, 63] [7, 14, 16, 24, 37, 43, 46, 47, 51, 61] [1, 3, 8, 11, 32, 33, 36, 38, 60, 62] [0, 2, 9, 10, 39, 45, 48, 49, 53, 63] [10, 19, 21, 29, 32, 34, 35, 38, 40, 62] [11, 18, 20, 28, 33, 41, 47, 50, 51, 55] [5, 7, 12, 15, 32, 34, 36, 37, 40, 42] [4, 6, 13, 14, 35, 43, 49, 52, 53, 57] [1, 14, 23, 25, 34, 36, 38, 39, 42, 44] [0, 15, 22, 24, 37, 45, 51, 54, 55, 59] [9, 11, 16, 19, 36, 38, 40, 41, 44, 46] [8, 10, 17, 18, 39, 47, 53, 56, 57, 61] [5, 18, 27, 29, 38, 40, 42, 43, 46, 48] [4, 19, 26, 28, 41, 49, 55, 58, 59, 63] [13, 15, 20, 23, 40, 42, 44, 45, 48, 50] [12, 14, 21, 22, 33, 43, 51, 57, 60, 61] [1, 9, 22, 31, 42, 44, 46, 47, 50, 52] [0, 8, 23, 30, 35, 45, 53, 59, 62, 63] [17, 19, 24, 27, 44, 46, 48, 49, 52, 54] [16, 18, 25, 26, 32, 33, 37, 47, 55, 61] [3, 5, 13, 26, 46, 48, 50, 51, 54, 56] [2, 4, 12, 27, 34, 35, 39, 49, 57, 63] [21, 23, 28, 31, 48, 50, 52, 53, 56, 58] [20, 22, 29, 30, 33, 36, 37, 41, 51, 59]
H_Z (32 checks, sparse supports)
[4, 6, 10, 12, 14, 27, 34, 43, 49, 57] [5, 10, 13, 23, 27, 31, 35, 42, 48, 56] [6, 8, 12, 14, 16, 29, 35, 36, 43, 61] [1, 7, 12, 15, 25, 29, 34, 37, 42, 60] [8, 10, 14, 16, 18, 31, 38, 47, 53, 61] [3, 9, 14, 17, 27, 31, 39, 46, 52, 60] [1, 10, 12, 16, 18, 20, 33, 39, 40, 47] [1, 5, 11, 16, 19, 29, 32, 38, 41, 46] [3, 12, 14, 18, 20, 22, 33, 42, 51, 57] [3, 7, 13, 18, 21, 31, 32, 43, 50, 56] [5, 14, 16, 20, 22, 24, 37, 43, 44, 51] [1, 5, 9, 15, 20, 23, 36, 42, 45, 50] [7, 16, 18, 22, 24, 26, 37, 46, 55, 61] [3, 7, 11, 17, 22, 25, 36, 47, 54, 60] [9, 18, 20, 24, 26, 28, 41, 47, 48, 55] [5, 9, 13, 19, 24, 27, 40, 46, 49, 54] [11, 20, 22, 26, 28, 30, 33, 41, 50, 59] [7, 11, 15, 21, 26, 29, 32, 40, 51, 58] [0, 13, 22, 24, 28, 30, 45, 51, 52, 59] [9, 13, 17, 23, 28, 31, 44, 50, 53, 58] [0, 2, 15, 24, 26, 30, 37, 45, 54, 63] [1, 11, 15, 19, 25, 30, 36, 44, 55, 62] [0, 2, 4, 17, 26, 28, 49, 55, 56, 63] [0, 3, 13, 17, 21, 27, 48, 54, 57, 62] [2, 4, 6, 19, 28, 30, 35, 41, 49, 58] [2, 5, 15, 19, 23, 29, 34, 40, 48, 59] [0, 4, 6, 8, 21, 30, 35, 53, 59, 60] [4, 7, 17, 21, 25, 31, 34, 52, 58, 61] [0, 2, 6, 8, 10, 23, 39, 45, 53, 62] [1, 6, 9, 19, 23, 27, 38, 44, 52, 63] [2, 4, 8, 10, 12, 25, 32, 39, 57, 63] [3, 8, 11, 21, 25, 29, 33, 38, 56, 62]
Code ID 64-16-6 · download JSON · raw on GitHub