Let {mathbb {Z}}_n = {Z_1, ldots , Z_n} be a design; that is, a collection of n points Z_j in [-1,1]^d. We study the quality of quantisation of [-1,1]^d by the points of {mathbb {Z}}_n and the problem of quality of coverage of [-1,1]^d by {{{mathcal {B}}}}_d({mathbb {Z}}_n,r), the union of balls centred at Z_j in {mathbb {Z}}_n. We concentrate on the cases where the dimension d is not small, dge 5, and n is not too large, nle 2^d. We define the design {{mathbb {D}}_{n,delta }} as a 2^{d-1} design defined on vertices of the cube [-delta ,delta ]^d, 0le delta le 1. For this design, we derive a closed-form expression for the quantisation error and very accurate approximations for the coverage area {text {vol}}{([-1,1]^d cap {{{mathcal {B}}}}_d({mathbb {Z}}_n,r))}. We provide results of a large-scale numerical investigation confirming the accuracy of the developed approximations and the efficiency of the designs {{mathbb {D}}_{n,delta }}.