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**Some results on \(2^{n-k}\) fractional factorial designs and search for minimum aberration designs.**
*(English)*
Zbl 0770.62063

As is well-known, a \(2^{n-k}\) fractional factorial design is determined by \(k\) generating words which in turn determine the defining contrasts subgroup. Associated with a \(2^{n-k}\) design is the wordlength pattern of the words in the underlying subgroup. The wordlength pattern of a design \(d\) is given by \(w(d)=(A_ 1(d),A_ 2(d),\dots)\), where \(A_ i(d)\) is the number of words of length \(i\). The smallest \(r\) such that \(A_ r(d)\neq 0\) is called the resolution of \(d\). If \(d_ 1\) and \(d_ 2\) are two \(2^{n-k}\) fractional factorial designs and \(r\) is the smallest \(i\) such that \(A_ i(d_ 1)\neq A_ i(d_ 2)\) then \(d_ 1\) has less aberration than \(d_ 2\) if \(A_ r(d_ 1)<A_ r(d_ 2)\). A design has minimum aberration if no other design has less aberration.

Among various results established in the paper the following are important:

(i) An upper bound is given for the length of the longest word in the defining contrasts subgroup; (ii) Minimum abberation \(2^{n-k}\) designs are presented for \(k=5\) and any \(n\); (iii) A method is given to test the equivalence of \(2^{n-k}\) designs; (iv) It is shown that minimum aberration \(2^{n-k}\) designs are unique for \(k\leq 4\).

Among various results established in the paper the following are important:

(i) An upper bound is given for the length of the longest word in the defining contrasts subgroup; (ii) Minimum abberation \(2^{n-k}\) designs are presented for \(k=5\) and any \(n\); (iii) A method is given to test the equivalence of \(2^{n-k}\) designs; (iv) It is shown that minimum aberration \(2^{n-k}\) designs are unique for \(k\leq 4\).

Reviewer: B.L.Raktoe (Kingston)

### MSC:

62K15 | Factorial statistical designs |

62K05 | Optimal statistical designs |

90C90 | Applications of mathematical programming |