Constructing cubature formulae of degree 5 with few points

This paper is devoted to construct a family of fifth degree cubature formulae for n-cube with symmetric measure and n-dimensional spherically symmetrical region. The formula forn-cube contains at most n^2+5n+3 points and for n-dimensional spherically symmetrical region contains only n^2+3n+3 points. Moreover, the numbers can be reduced to n^2+3n+1 and n^2+n+1 if n=7 respectively, the latter of which is minimal.

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