= Solution
For this fraction, $SCW=-1$ and $SCMT=1$, so their product is $WMT=-1$. The signed defining relation is
$$
I=-SCW=SCMT=-WMT.
$$
The complete <factorial contrast> alias classes are
$$
\begin{aligned}
S&=-CW=CMT=-SWMT,\\
C&=-SW=SMT=-CWMT,\\
W&=-SC=SCWMT=-MT,\\
M&=-SCWM=SCT=-WT,\\
T&=-SCWT=SCM=-WM,\\
ST&=-CWT=CM=-SWM.
\end{aligned}
$$
The 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 \b[design (b) is suitable under the stated interaction assumptions]. Its intercept and six requested columns are mutually orthogonal, with $X^TX=8I_7$, so each coded regression coefficient is $\widehat\beta_A=8^{-1}\sum_{i=1}^8\chi_A(x_i)Y_i$. The corresponding high-minus-low <main effect> is $2\widehat\beta_A$.
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.
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