Perfect Codes in Euclidean Lattices: Bounds and Case Studies
Giselle Ribeiro de Azeredo Silva Strey, Antonio C. de A. Campello, João E. Strapasson, Sueli I. R. Costa · 2019
In the present paper, we investigate the existence of lattice perfect codes when considered as sublattices of other lattices under the Euclidean metric. We generalize bounds on the radius of perfect codes in a generic lattice, previously known for the cubic lattice. The new bounds are based on covering density, and covering radius of the ambient lattices, and, along with algebraic methods, allow to characterise all perfect codes in small dimension for a given ambient lattice. We provide case studies for some well known ambient lattices, such as the hexagonal lattice, and the checkerboard lattices. In contrast to the cubic lattice, these case studies show that, by changing the ambient lattice, one can find rich sets of perfect codes.