A Four-Dimensional Toric Code on the D4 Lattice: [[16L4,6,2L2]] with Weight-4 X-Stabilizers

We study a four-dimensional toric code on the D4 lattice, the densest four-dimensional lattice packing. Qubits sit on the 16L4 triangles of its Delaunay decomposition into regular 16-cells. Each X-stabilizer acts on the four triangles of a tetrahedron, the smallest possible weight, and each Z-stabilizer on the eight triangles meeting along an edge. Up to exchanging X and Z, this is the face code of the dual 24-cell honeycomb, which has been proposed before without its parameters. We prove that the code is [[16L4,6,2L2]] for every even L≥4, with dX=2L2 and dZ=4L2 exactly. A logical operator must cast a shadow covering a two-dimensional cross-section of the torus, and a single triangle shadows an area of at most 1/2; explicit operators built from flat planes attain the resulting bounds in both sectors. The code reaches distance d with 4d2 qubits against 6d2 for the standard hypercubic code, although rotated hypercubic codes need fewer. Hook errors from the weight-8 checks cost no more than those from the weight-4 checks. Excitations in both sectors are loops, and the slices of the lattice are face-centered cubic. Published scripts verify all results.