An Imbedded Family of Fully Symmetric Numerical Integration Rules

We consider an imbedded family of $m + 1$ generator $2m + 1$ degree fully symmetric rules for numerical integration over an n-dimensional hypercube. The rules are shown to exist in theory for arbitrary degree. In practice, the rules are most economical and useful for $3 \leqq m \leqq 6$, $3 \leqq n \leqq 3m$.