Example 10.4.1. An engineer used the following eight-run design in five factors:

The columns \(A\), \(B\), \(C\) form a full \(2^3\) in standard order, so they are the basic factors. Comparing \(D\) with the products of these columns, \(D\) agrees with \(BC\) in every row (row \(1\): \((-)(-) = +\); row \(3\): \((+)(-) = -\); and so on), and \(E\) agrees with \(ABC\). So the generators are \(D = BC\) and \(E = ABC\), that is \[ I = BCD = ABCE , \qquad \text {and their product}\qquad I = ADE . \] The defining relation is \(I = BCD = ABCE = ADE\), and the design is of resolution III. The alias structure is

No \(2^{5-2}\) design has resolution above III: if two generator words both had four or more of the five letters, they would share at least three, and their product would have at most two. Every \(2^{5-2}_{\mathrm {III}}\) design has at least two words of length three, and this one has exactly two, so in the sense of minimum aberration it is as good as any.