Combined-optimal designs (Li and Lin, 2003) are obviously the best choices for the initial designs if we partition the experiment into two parts with equal size to obtain some information about the process, especially for the case not considerign the blocking factor. In this pape, the definition of combined-optimal design is extended to the case when blocking factor is significant, and this new class of designs is called blocded combined-optimal designs. Some general results are obtained which relate 2k-pIII inital designs with their complementary designs when k>5n8, where n=2k-p. By applying these results, we are able to characterize 2k-pIII combined-optimal designs or blocked combined-optimal dwdigns in terms of their complementary designs. It is also proved that both 2k-pIII compbined-optimal and blocked combined-optimal designs are not minimum aberration designs when k>5n8 and n-1-k>2. And some combined-optimal and bloked combined -optimal designs with 16 and 32 runs are constructed fo rillustration.

In this paper, by introducing a concept of consulting design and based on the connection between factorial design theory and coding theory, we obtain combinatorial identities that relate the wordlength pattern of a regular mixed factorial design to that of its consulting design. According to these identities, we further-more establish the general and unified rules for idejntifying mini-mum aberration mixed factorial designs through their consulting designs. It is an improvement and generalization of the results in Mukerjee and Wu (2001).

Chen and Cheng (Ann. Statist. 27 (1999) 1948) got the relationships between a blocked symmetrical factorial design and its residual design. In this paper, we extend the study to the case of blocked mixed-level factorial designs. By introducing a concept of blocked consulting design and using MacWilliams identities and Krawtchouk polynomials in coding theory, we obtain combinatorial identities that govern the relationships between the partitioned wordlength patterns of a blocked regular mixed factorial design and that of its blocked consulting design. Based on these identities, we furthermore establish some general rules for identifying minimum aberration blocked mixed factorial designs in terms of their blocked consulting designs. As applications, some blocked 412n and 422n designs with 16 and 32 runs are tabulated.

For a fixed number of runs, when can designs have clear main effects or clear two-factor interactions (in brief, 2fi' s)? This paper gives the maximum value of n in sn m designs containing clear main effects or clear 2fi' s, where s is any prime or prime power. It is a generalization of the related results in Chen and Hedayat (J. Statist. Plann. Inference 75 (1998) 147-158) for two-level designs. It is further concluded that the weak minimum aberration designs have a maximum number of clear main effects for two-level designs. A collection of designs containing most clear main effects or clear 2fi' s for 16, 32, 27, and 81 runs is given.

In this paper formulas for computing the E (s2) values of some kinds of E (s2) optimal super-saturated designs are given, and a general algorithm of constructing E (s2) optimal supersaturated designs from cyclic BIBDs is proposed. Within this class of designs, by further discriminating the pairwise correlations, e cient designs of runs from 6 to 24 are constructed and tabulated. Comparisons with other existing designs are made at last, demonstrating the e ectiveness of our method.