Both the clear effects and minimum aberration criteria are the important rules for the design selection. In this paper, it is proved that some 2IVm−p designs have weak minimum aberration, by considering the number of clear two-factor interactions in the designs. And some conditions are provided, under which a 2IVm−p design can have the maximum number of clear two-factor interactions and weak minimum aberration at the same time. Some weak minimum aberration 2IVm−p designs are provided for illustrations and two nonisomorphic weak minimum aberration 2IV13−6 designs are constructed at the end of this paper.

This paper is concerned with the uniformity of a certain kind of resolvable incomplete block (RIB for simplicity) design which is called the PRIB design here. A sufficient and necessary condition is obtained, under which a PRIB design is the most uniform in the sense of a discrete discrepancy measure, and the uniform PRIB design is shown to be connected. A construction method for such designs via a kind of U-type designs is proposed, and an existence result of these designs is given. This method sets up an important bridge between PRIB designs and U-type designs.

A uniform design seeks design points that axe uniformly scattared on the experimental domain. It has been popular since 1980. In this paper, we employ the discrete discrepancy as the measure of uniformity for constructing uniform designs. The equivalency among the discrete discrepancy, non-orthogonal measure E (fNoo) and minimum generalized aberration (MGA) for comparing U-type designs U(n; qm) is given. The link between the two apparently unrelated areas of uniform designs and super-simple resolvable L-designs in combinatorial design theory is shown. Through super-simple resolvable t-designs, under the discrete discrepancy, a new method is proposed for constructing uniform designs Un (q,m), in which any of the q2 levelcombinations between any two distinct columns can appear at most twice and any two columns are not fully aliased. Many new uniform designs are obtained.

Supersaturated design is a form of fractional factorial design and has recently received much interest because of its potential in factor screening experiments. This paper mainly concerns the existing criteria for mixed-level designs, i.e. the x2(D) criterion (for design D) proposed by Yamada and Matsui (J. Statist.Plann. Inference 104 (2002) 459), the E (fNOD) criterion proposed by Fang et al. (Metrika 58 (2003b) 279) and the minimum moment aberration (MMA) proposed by Xu (Statist. Sinica 13 (2003) 691). A lower bound of x2(D) is obtained along with the sufficient and necessary condition for achieving it. The connections between x2(D) and other criteria are discussed. Especially, it is shown that x2(D) and the second power moment x2(D) are in fact equivalent to each other and hence the two criteria, x2(D) and MMA, share the same optimal supersaturated designs, which can be constructed from saturated orthogonal arrays and other supersaturated designs.

Incomplete block designs are a type of experimental design layout that has had widespread use in science and engineering. A balanced incomplete block design (BIB) can be characterized by the balanced arrangement of its design points. In this article, incomplete block designs will be investigated from a new perspective: the perspective of uniformity in distribution of design points. A design is of high uniformity, or low discrepancy, if its design points distribute uniformly over the entire design space. The authors use a general discrepancy measure to prove theoretically that BIBs are the most uniform ones among all binary incomplete block designs.