Abstract. A common problem that experimenters face is the choice of fractional fac-torial designs. Minimum aberration designs are commonly used in practice. There are situations in which other designs meet practical needs better. A catalogue of designs would help experimenters choose the best design. Based on coding theory, new meth-ods are proposed to classify and rank fractional factorial designs efficiently. We have completely enumerated all 27 and 81-run designs, 243-run designs of resolution IV or higher, and 729-run designs of resolution V or higher. A collection of useful fractional factorial designs with 27, 81, 243 and 729 runs is given. This extends the work of Chen, Sun and Wu (1993), who gave a collection of fractional factorial designs with 16, 27, 32 and 64 runs. Key words: clear effect, linear code, minimum aberration, moment aberration, reso-lution.

of doubling for constructing two-level fractional factorial designs. They showed that for 9N/32 ≤ n ≤ 5N/16, all minimum aberration designs with N runs and n factors are projections of the maximal design with 5N/16 factors which is constructed by repeatedly doubling the 2 5−1 design defined by I = ABCDE. This paper develops a general complementary design theory for doubling. For any design obtained by repeated doubling, general identities are established to link the wordlength patterns of each pair of complementary projection designs. A rule is developed for choosing minimum aberration projection designs from the maximal design with 5N/16 factors. It is further shown that for 17N/64 ≤ n ≤ 5N/16, all minimum aberration designs with N runs and n factors are projections of the maximal design with N runs and 5N/16 factors. 1. Introduction. Fractional

This paper considers the construction of minimum aberration (MA) blocked factorial designs. Based on coding theory, the concept of minimum moment aberration due to Xu [Statist. Sinica 13 (2003) 691–708] for unblocked designs is extended to blocked designs. The coding theory approach studies designs in a row-wise fashion and therefore links blocked designs with nonregular and supersaturated designs. A lower bound on blocked wordlength pattern is established. It is shown that a blocked design has MA if it originates from an unblocked MA design and achieves the lower bound. It is also shown that a regular design can be partitioned into maximal blocks if and only if it contains a row without zeros. Sufficient conditions are given for constructing MA blocked designs from unblocked MA designs. The theory is then applied to construct MA blocked designs for all 32 runs, 64 runs up to 32 factors, and all 81 runs with respect to four combined wordlength patterns.

A supersaturated design is a design whose run size is not large enough for estimating all the main effects. The goodness of multi-level supersaturated designs can be judged by the generalized minimum aberration criterion proposed by Xu and Wu [Ann. Statist. 29 (2001) 1066–1077]. A new lower bound is derived and general construction methods are proposed for multi-level supersaturated designs. Inspired by the Addelman–Kempthorne construction of orthogonal arrays, several classes of optimal multi-level supersaturated designs are given in explicit form: Columns are labeled with linear or quadratic polynomials and rows are points over a finite field. Additive characters are used to study the properties of resulting designs. Some small optimal supersaturated designs of 3, 4 and 5 levels are listed with their properties. 1. Introduction. As science and technology have advanced to a higher level, investigators are becoming more interested in and capable of studying large-scale systems. Typically these systems have many factors that can be varied during design

Fractional factorial designs are widely used in practice and typically chosen according to the minimum aberration criterion. A sequential algorithm is developed for constructing efficient fractional factorial designs. A construction procedure is proposed that only allows a design to be constructed from its minimum aberration projection in the sequential build-up process. To efficiently identify nonisomorphic designs, designs are divided into different categories ac-cording to their moment projection patterns. A fast isomorphism check procedure is developed by matching the factors using their delete-one-factor projections. A method is proposed for constructing minimum aberration designs using only a partial catalog of some good designs. Minimum aberration designs are constructed for 128 runs up to 64 factors, 256 runs up to 28 factors, and 512, 1024, 2048, and 4096 runs up to 23 or 24 factors. Furthermore, this algorithm is used to completely enumerate all 128-run designs of resolution 4 up to 30 factors, all 256-run designs of resolution 4 up to 17 factors, all 512-run designs of resolution 5, all 1024-run designs of resolution 6, and all 2048- and 4096-run designs of resolution 7.

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The research of developing a general methodology for the construction of good nonregular designs has been very active in the last decade. Recent research by Xu and Wong [Statist. Sinica 17 (2007) 1191–1213] suggested a new class of nonregular designs constructed from quaternary codes. This paper explores the properties and uses of quaternary codes toward the construction of quarter-fraction nonregular designs. Some theoretical results are obtained regarding the aliasing structure of such designs. Optimal designs are constructed under the maximum resolution, minimum aberration and maximum projectivity criteria. These designs often have larger generalized resolution and larger projectivity than regular designs of the same size. It is further shown that some of these designs have generalized minimum aberration and maximum projectivity among all possible designs.

A criterion of design efficiency, under model uncertainty, is studied with reference to possibly nonregular fractions of general factorials. The criterion is expressed in terms of the departure of the design from being an orthogonal array of strengths three or four. A Kronecker calculus for factorial arrangements facilitates the derivation. The results are followed up with a numerical study and the findings are compared with those based on other design criteria.

This paper aims to generalize and unify classical criteria for comparisons of balanced lattice designs, including fractional factorial designs, supersaturated designs and uniform designs. We present a general majorization framework for assessing designs, which includes a stringent criterion of majorization via pairwise coincidences and flexible surrogates via convex functions. Classical orthogonality, aberration and uniformity criteria are unified by choosing combinatorial and exponential kernels. A construction method is also sketched out.