2.4 Fractional $3^{m-k}$ design

Definition 2.4.1   A fractional $3^{m-k}$ design addresses $m$ factors, each of them having three levels. The mesure points are selected to provide a full factorial design for each combination of $m-k$ factors.

Example 2.4.2   Here is a $3^{3-1}$ design of $9$ measure points addressing three ternary factors, each of them being encoded in $\left\{ 0,1,2\right\}$ :

$$\begin{array}{cccccccccc} \char93 &1&2&3&4&5&6&7&8&9\cr A&0&0... ...2&2&2\cr B&0&1&2&2&0&1&1&2&0\cr C&0&1&2&0&1&2&0&1&2 \end{array}$$

Exercise 2.4.3   Check that the previous design is a full design for each triple of factors.

Exercise 2.4.4   Build a $3^{4-2}$ design of $9$ measure points adressing $4$ factors. Determine its eigenvalues.

Exercise 2.4.5   Build a $3^{4-1}$ design of $27$ measure points adressing $4$ factors. Determine its eigenvalues. Which are the interactions that can be detected by this design ?