Theory of optimal blocking of 2ⁿ⁻ᵐ designs

In this paper, we define the blocking wordlength pattern of a blocked fractional factorial design by combining the wordlength patterns of treatment-defining words and block-defining words. The concept of minimum aberration can be defined in terms of the blocking wordlength pattern and provides a good measure of the estimation capacity of a blocked fractional factorial design. By blending techniques of coding theory and finite projective geometry, we obtain combinatorial identities that govern the relationship between the blocking wordlength pattern of a blocked 2ⁿ⁻ᵐ design and the split wordlength pattern of its blocked residual design. Based on these identities, we establish general rules for identifying minimum aberration blocked 2ⁿ⁻ᵐ designs in terms of their blocked residual designs. Using these rules, we study the structures of some blocked 2ⁿ⁻ᵐ designs with minimum aberration.