Algebraic lattices via polynomial rings

Abstract

Signal constellations having lattice structure have been studied as meaningful means for signal transmission over Gaussian channel. Usually the problem of finding good signal constellations for a Gaussian channel is associated with the search for lattices with high packing density, where in general the packing density is usually hard to estimate. The aim of this paper was to illustrate the fact that the polynomial ring $\mathbb {Z}[x]$ can produce lattices with maximum achievable center density, where $\mathbb {Z}$ is the ring of rational integers. Essentially, the method consists of constructing a generator matrix from a quotient ring of $\mathbb {Z}[x]$.