$A$-Optimal Weighing Designs when $N \equiv 3 (\operatorname{mod} 4)$

Optimal experimental designs for fMRI via circulant biased weighing designs

“…With these facts and some algebra similar to that in Sathe and Shenoy (1989), we can see that, for |c| > 3,…”

“…Our claim in Theorem 2.2 then follows from (A.1), and Corollary 3.3 of Cheng (1987). This approach is similar to that of Sathe and Shenoy (1989), and Galil and Kiefer (1980), where weighing designs under the unbiasedness assumption are considered. We now present the details of our proof.…”

“…In Theorem 2.2, we provide an N 0 (K, 1) for a design to be Φ 1 -optimal (i.e., A-optimal). Our approach for deriving this bound for N is analogous to that of Galil and Kiefer (1980), and Sathe and Shenoy (1989). A proof is provided in the Appendix.…”

“…With these results, we can now work on B 0,s ⊂ B s that consists of block matrices having m(< K) block sizes with r 1 − 1 = · · · = r v − 1 = r v+1 = · · · = r m = r ≥ 1, and v ≥ 1. We note that, for any m 0 ≥ 1 and r 0 ≥ 2, a block matrix having (m, v, r) = (m 0 , 0, r 0 ) can be treated as a block matrix with (m, v, r) = (m 0 , m 0 , r 0 − 1), and is thus in B 0,s ; see also Sathe and Shenoy (1989). Consequently, the only block matrix in B s that is left out from B 0,s is B * = (N + 1)I K − J K , which has (m, v, r) = (K, 0, 1), or equivalently, (m, v, r) = (K, K, 0).…”

“…Lemma A.4 then follows from Lemma A.3. To that end, we need the following lemmas, which are extensions of results in Sathe and Shenoy (1989). Lemma A.5 is a well-known result, and the proof is omitted.…”