  Given a sphere of any radius $r$ in an $n$-dimensional Euclidean space, we
study the coverings of this sphere with solid spheres of radius one. Our goal
is to design a covering of the lowest covering density, which defines the
average number of solid spheres covering a point in a bigger sphere. For a
growing dimension $n,$ we design a covering that has covering density of order
$(n\ln n)/2$ for the full Euclidean space or for a sphere of any radius $r>1.$
This new upper bound reduces two times the asymptotic order of $n\ln n$
established in the classical Rogers bound.
