Constructing cubature formulae of degree 5 with few points

This paper is devoted to construct a family of fifth degree cubature formulae for nn-cube with symmetric measure and nn-dimensional spherically symmetrical region. The formula fornn-cube contains at most n2+5n+3n2+5n+3 points and for nn-dimensional spherically symmetrical region contains only n2+3n+3n2+3n+3 points. Moreover, the numbers can be reduced to n2+3n+1n2+3n+1 and n2+n+1n2+n+1 if n=7n=7 respectively, the latter of which is minimal.