β-wordlength Pattern Under The Complex Contrasts And Its Calculation

The minimum aberration criterion is an important criterion evaluating fractional factorial designs with qualitative factors. When a design contains quantitative factors, a polynomial model is often established to make statistical inference and two combinatorial isomorphism designs may have different statistical inference ability. However, these two designs cannot be distinguished by minimum aberration criterion. Using the indicator function based on the orthogonal polynomial basis, Cheng and Ye(2004) proposed the concept of β-wordlength pattern, which can evaluate fractional factorial designs with quantitative factors. They pointed out that a good design should have a smaller β-wordlength pattern. But we cannot establish direct connection between β-wordlength pattern and defining contrast subgroup. Also the calculation of β-wordlength pattern based on its definition is complicated. Therefore, this article aims to make an improvement to β-wordlength pattern.First we establish the form of L1 weight distribution of the dual code about the weight composition according to the Mac Williams identity in coding theory. Then, based on the indicator function, we proposed the concept of β-wordlength pattern under the complex contrasts and give some properties. Through the relationship between β-wordlength pattern under the complex contrasts and defining contrast subgroup, we establish the relationship between β-wordlength pattern under the complex contrasts of q-level designs and L1 weight distribution of its dual code. Finally, we give the calculation formula of β-wordlength pattern under the complex contrasts for 3n-m designs base on the weight composition.