Description
In this work a lattice means a discrete and infinite regular arrangement of points in n-dimensional Euclidean space. Lattices are associated to long-standing open problems in mathematics such as the sphere packing and sphere covering problems. Although those are problems of a geometric nature, lattices with “interesting” features have been constructed from inherently algebraic structures such as groups, error-correcting codes, and number fields. Lattices of high packing density have long been used in field of telecommunications. More specifically, “dense” lattices can be used to construct signal sets for transmitting information accurately at high rates. In this work we show that polynomials with integer coefficients can be used to construct two, three, and four-dimensional lattices of maximum achievable packing density. In fact, families of such lattices in each of those dimensions are produced. When the roots of the polynomials are all real, lattices of maximum diversity are obtained. Besides being attractive for use over the Gaussian channel, those lattices are also attractive for use in certain mobile fading channels.
