Construction of blocked designs with multi block variables

When experimental units are inhomogeneous, blocking the experimental units into categories is crucial so as to estimate the treatment effects precisely. In practice, the inhomogeneity often comes from different sources known as block variables in design terminology. The paper considers the blocking problems with multi block variables. The construction methods of the optimal blocked regular $ 2^{n-m} $ designs with multi block variables under the general minimum lower order confounding criterion for $ \frac{5N}{16}+1\leq n \leq N-1 $ are provided, where $ N = 2^{n-m} $.