On second order orthogonal Latin hypercube designs

In polynomial regression modeling, the use of a second order orthogonal Latin hypercube design guarantees that the estimates of the first order effects are uncorrelated to each other as well as to the estimates of the second order effects. In this paper, we prove that such designs with n runs and k>2 columns do not exist if n≡4mod8. Furthermore, we prove that second order Latin hypercube designs with k≥4 columns that guarantee the orthogonality of two-factor interactions which do not share a common factor, do not exist for even n that is not a multiple of 16. Finally, we investigate the class of symmetric orthogonal Latin hypercube designs (SOLHD), which are a special subset of second order orthogonal Latin hypercube designs. We describe construction techniques for SOLHDs with n runs and (a) k≤4 columns, when n≡0mod8 or n≡1mod8, (b) k≤8 columns when n≡0mod16 or n≡1mod16 and (c) k=4 columns when n≡0mod16 or n≡1mod16, that guarantee the orthogonality of two-factor interactions which do not share a common factor. Finally, we construct and enumerate all non-isomorphic SOLHDs with n≤17 runs and k≥2 columns, as well as with 19≤n≤20 runs and k=2 columns.