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In this contribution to the theory of lattice rules for multidimensional numerical integration, we first establish bounds for various efficiency measures which lead to the conclusion that in the search for efficient lattice rules one should concentrate on lattice rules with large first invariant. Then we prove an existence theorem for efficient lattice rules of rank 2 with prescribed invariants, which extends an earlier result of the author for lattice rules of rank 1.

The existence of efficient lattice rules for multidimensional numerical integration