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We consider an n-dimensional integration rule Rt1") of degree 2t - I and of hypercubic symmetry. We derive theorems which set a lower bound in terms of n and t on the number of function evaluations such a rule requires. These results apply to spaces of integration which have hypercubic symmetry. In certain cases this bound is very close to the number of points required by a known rule.

Limits on the number of function evaluations required by certain high-dimensional integration rules of hypercubic symmetry