Constructing Cubature Formulas of Degree 5 with Few Points

Abstract: This paper will devote to construct a family of fifth degree cubature formulae for $n$-cube with symmetric measure and $n$-dimensional spherically symmetrical region. The formula for $n$-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 later of which is minimal.