The index of merit of kth-copy integration lattices

r2d2lri is an automatic two-dimensional cubature algorithm that demonstrates the practical value of using an augmentation sequence consisting of (2k)2-copy lattices as a basis for numerical integration. This paper investigates use of similar embedded augmentation sequences in higher dimensions by developing theoretical results relating to the index of merit of s-dimensional (2k)s-copy lattices generated from rank-1 simple lattices. The theoretical results can be used to guide the search for good augmentation sequences in s dimensions in the sense of high index of merit.