We give an explicit upper bound of the minimal number _T,n of balls of radius 1/2 which form a covering of a ball of radius T > 1/2 in ⁿ, n \geq 2. The asymptotic estimates of _T,n we deduce when n is large are improved further by recent results of Böröczky, Jr. and Wintsche on the asymptotic estimates of the minimal numberof equal balls of ⁿ covering the sphere S^n-1. The optimality of the asymptotic estimates is discussed.