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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 $$k\leq 4$$.

##### MSC:
 62K15 Factorial statistical designs 62K05 Optimal statistical designs 90C90 Applications of mathematical programming
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