“…We also show some upper bounds on G n that were obtained in [3], [5] by evaluating the mean squared error of certain lattices when used to quantize uniformly distributed inputs. The following lattices were used: the so-called Voronoi lattices of the first type, A n * (the subscript indicates the dimension, and the asterisk that these are the duals of the root lattices A n -see [3]); the lattices D n * ; and, in the column of the It is known that A 1 * (the integers) and A 2 * (the familiar two-dimensional hexagonal lattice) are optimal for quantizing uniformly distributed inputs in one and two dimensions ([8], [10], [18]), and that the body-centered cubic lattice A 3 * is similarly optimal among lattice quantizers in three dimensions [1]. It can be seen from the Gosset's (see [4], [5], [15], [16], [19], [24]), and the existence of these two lattices explains the small gaps in the [14]; see also [21].…”