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**1 - 4**of**4**### 2-level fractional factorial designs which are the union of non trivial regular designs

"... Abstract Every fraction of a 2-level factorial design is a union of points, each of them being trivially a regular fraction. In order to find non-trivial decomposition, we derive a condition for the inclusion of a regular fraction in a generic fraction as follows. Regular fractions are characterize ..."

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Abstract Every fraction of a 2-level factorial design is a union of points, each of them being trivially a regular fraction. In order to find non-trivial decomposition, we derive a condition for the inclusion of a regular fraction in a generic fraction as follows. Regular fractions are characterized by a polynomial indicator function of the form +1} (Fontana et al., 2000). A regular fraction R is a subset of a given fraction The practical applicability of the previous condition is discussed in the second part of the paper.

### 2-LEVEL FRACTIONAL FACTORIAL DESIGNS WHICH ARE THE UNION OF NON TRIVIAL REGULAR DESIGNS

, 710

"... Abstract. Every fraction is a union of points, which are trivial regular fractions. To characterize non trivial decomposition, we derive a P condition for the inclusion of a regular fraction as follows. Let F = α bαXα be the indicator polynomial of a generic fraction, see Fontana et al, JSPI 2000, 1 ..."

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Abstract. Every fraction is a union of points, which are trivial regular fractions. To characterize non trivial decomposition, we derive a P condition for the inclusion of a regular fraction as follows. Let F = α bαXα be the indicator polynomial of a generic fraction, see Fontana et al, JSPI 2000, 149-172. Regular fractions are characterized by R = P 1 l α∈L eαXα, where α ↦ → eα is an group homeomorphism from L ⊂ Z d 2 into {−1, +1}. The regular R is a subset of the fraction F if FR = R, which in turn is equivalent to P t {α1... αk} is a generating set of L, and R = 1 F(t)R(t) =

### Generation of Fractional Factorial Designs

, 906

"... The joint use of counting functions, Hilbert basis and Markov basis allows to define a procedure to generate all the fractions that satisfy a given set of constraints in terms of orthogonality. The general case of mixed level designs, without restrictions on the number of levels of each factor (like ..."

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The joint use of counting functions, Hilbert basis and Markov basis allows to define a procedure to generate all the fractions that satisfy a given set of constraints in terms of orthogonality. The general case of mixed level designs, without restrictions on the number of levels of each factor (like primes or power of primes) is studied. This new methodology has been experimented on some significant classes of fractional factorial designs, including mixed level orthogonal arrays.