 Solution

For this fraction, $SCW=-1$ and $SCMT=1$, so their product is $WMT=-1$. The signed defining relation isThe complete factorial contrast alias classes areThe five main effects and $ST$ occupy six distinct alias classes, and none aliases with the intercept. Every undesired two-factor term in these classes is assumed negligible, and all terms of order at least three are also assumed negligible. Thus design (b) is suitable under the stated interaction assumptions. Its intercept and six requested columns are mutually orthogonal, with $X^{T}X=8I_7$, so each coded regression coefficient is ${\hat{\beta }}_A=8^{-1}{\sum }_{i=1}^8{\chi }_A(x_i)Y_i$. The corresponding high-minus-low main effect is $2{\hat{\beta }}_A$.

There is one residual degree of freedom, but no replicated setting and therefore no separate pure error estimate. Suitability here means estimability under the stated model; it does not protect estimates from active interactions that the assumptions exclude.