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Functional magnetic resonance imaging (fMRI) is a pioneering technology for studying brain activity in response to mental stimuli. Although efficient designs on these fMRI experiments are important for rendering precise statistical inference on brain functions, they are not systematically constructed. Design with circulant property is crucial for estimating a hemodynamic response function (HRF) and discussing fMRI experimental optimality. In this paper, we develop a theory that not only successfully explains the structure of a circulant design, but also provides a method of constructing efficient fMRI designs systematically. We further provide a class of two-level circulant designs with good performance (statistically optimal), and they can be used to estimate the HRF of a stimulus type and study the comparison of two HRFs. Some efficient three- and four-levels circulant designs are also provided, and we proved the existence of a class of circulant orthogonal arrays.

Much contrary to popular belief and even a published result, it is seen that orthogonal array plus one run plans are not necessarily optimal, within the relevant class, for general $s_1 \times \dots \times s_m$ factorials. A broad sufficient condition on $s_1,\dots, s_m$ ensuring the optimality of such plans has been worked out. This condition covers, in particular, all symmetric factorials and thus strengthens some previous results.

We study the construction of orthogonal arrays with three levels and index 1 and the existence of orthogonal arrays with three levels and index 2. For strength greater than or equal to 2, we show that orthogonal arrays with three levels and index 1 are unique, and we establish the maximum number of factors for orthogonal arrays with three levels and index 2.

Orthogonal arrays with variable numbers of symbols are shown to be universally optimal as fractional factorial designs. The orthogonality of completely regular Youden hyperrectangles ($F$-hyperrectangles) is defined as a generalization of the orthogonality of Latin squares, Latin hypercubes, and $F$-squares. A set of mutually orthogonal $F$-hyperrectangles is seen to be a special kind of orthogonal array with variable numbers of symbols. Theorems on the existence of complete sets of mutually orthogonal $F$-hyperrectangles are established which unify and generalize earlier results on Latin squares, Latin hypercubes, and $F$-squares.

Rectangular lattice designs are shown to be generally balanced with respect to a particular decomposition of the treatment space. Efficiency factors are calculated, and the analysis, including recovery of interblock information, is outlined. The ideas are extended to rectangular lattice designs with an extra blocking factor.

Finney has used orthogonal partitions in the context of the search for higher order (coarser) partitions of given Latin squares. Hedayat and Seiden use the term $F$-square to denote higher order partitions that are orthogonal to both rows and columns. This note is a short expository treatment of orthogonal partitions in general and is based on the identification of a partition with a vector subspace of Euclidean $N$-space $R^N$. This identification is not new as it is part of the usual vector space approach to analysis of variance. This approach puts the concept of orthogonal partitions in a simple light unencumbered by the language of design of experiments. Another advantage is that certain published bounds on the maximum number of orthogonal partitions of specified type are immediate from the dimensionality restriction imposed by $R^N$. In addition, some counting problems are identified which are of possible interest to researchers in design of experiments and combinatorics.

For the minimum weight in $s$-ary $(n, p)$ linear codes (or for the maximum resolution in $s^{n-p}$ designs), an upper bound has been obtained by Plotkin [4] where $s$ is a prime power. The purpose of this note is to obtain an improvement of the Plotkin's upper bound. Main result is as follows: when $p \geqq 2$, the maximum resolution $R_p(n, s)$ of any $s^{n-p}$ design satisfies the following: \begin{equation*}\begin{split}R_p(n, s) = s^{p-1}q\quad\text{if} m = 0, 1 \\ &\leqq s^{p-1}q + \lbrack s^{p-2}(s - 1)(m - 1)/(s^{p-1} - 1)\rbrack \\ \text{if} m = 2, 3, \cdots, s^{p-1} \\ &\leqq s^{p-1}q + \lbrack(s - 1)m/s\rbrack\quad\text{if} m = s^{p-1} + 1, \cdots, N - 1,\end{split}\end{equation*} where $\lbrack x\rbrack$ is the greatest integer not exceeding $x, n = qN + m$ and $N = (s^p - 1)/(s - 1)$.

It is shown that Bush's bound for maximum number of constraints in an orthogonal array of index unity is uniformly better than Rao's bound. In addition it is shown, using an argument similar to that needed in the proof of the above result, that Noda's characterization of parameters in orthogonal arrays of strength 4 achieving equality in Rao's bound, leads easily to a similar characterization in arrays of strength 5. These results are useful designing experiments for quality control.

Consider the class of balanced fractional $2^m$ factorial designs of resolution VII. Within this class, optimal designs with respect to the trace criterion are given for any fixed $N$ assemblies, which satisfy (i) $m = 6, 42 \leqq N \leqq 64$, (ii) $m = 7, 64 \leqq N \leqq 90$ and (iii) $m = 8, 93 \leqq N \leqq 128$. The covariance matrices of the estimates of the effects are also given for such designs.

The main purpose of the paper is to obtain the best possible bound on the number of orthogonal F-squares with certain parameters and to give a construction method for some families of orthogonal F-squares which achieve this bound. Also, presented is a set of four mutually orthogonal F-squares of order 6 based on three symbols. This later design is important because there is no orthogonal Latin squares of order 6 which could be used for this purpose as has been pointed out by Hedayat and Seiden (1970). The authors indicate a method of composing orthogonal F-squares. It is shown under what condition a set of orthogonal F-squares can be transformed into an orthogonal array, a structure which is useful for factorial experimentation. (Modified author abstract)