Sphere-packings, lattices, and groups
Sphere-packings, lattices, and groups
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A lower bound of the form (\frac{2n}{n+1})^{\frac{1}{n}}γ_{n-1}^{\frac{n-1}{n}} is derived on the coding gainγ_n of the densest n-dimensional (n-D) lattice(s). The bound is obtained based on constructing ann-D lattice which consists of parallel layers. Each layer isselected as a translated version of a densest ( n-1)-D lattice.0The relative positioning of the layers is adjusted to make the coding gainas large as possible. For large values of n, the bound isimproved through tightening Ryskov‘s inequality on covering radius andminimum distance of a lattice.