Variance impact of the choice of η(·) in the k>2 equireplicate design construction

Develop a theoretical analysis that quantifies how the choice of the strictly increasing sequence η(·) used in Algorithm \ref{alg:k>2Design} affects block overlaps and thus the variance of incomplete U-statistics of order k>2 constructed via this algorithm.

Background

Algorithm \ref{alg:k>2Design} produces r-equireplicate designs for k>2 using cyclic permutations indexed by a strictly increasing sequence η(·). While the algorithm ensures linear-time construction and allows large design sizes, the variance of the resulting U-statistics depends on the degree of block overlaps.

The authors note that the choice of η(·) controls overlap structure and influences the variance, but they do not provide a quantitative analysis. A rigorous treatment would guide practitioners in selecting η(·) to optimize variance and dependence properties.