## Notes on Certain Constructions.

• Note (c): Complete factorial. Whenever N can be factorized as

N     =     s1 s2 . . . sa

where the si are relatively prime, there is a complete factorial OA(N, s11 s21 . . . sa1) consisting of all possible runs in which the first factor takes any level between 0 and s1-1, the second factor takes any level between 0 and s2-1, etc. (Of course this is simply the direct product of the appropriate number of trivial orthogonal arrays.)

• Note (h): Hadamard construction. If A is an n X n Hadamard matrix with entries +1, -1, and u is a single column

[0, 1, . . . , n - 1 ]tr,

then

 u A u -A

is an OA( 2n, n1 2n). The Hadamard matrices used can be found in the accompanying table.

• Note (j): Juxtaposition. There are two versions of this trivial construction.
1. Combining an OA(N', (s'1)1 s2k) and an OA(N'', (s''1)1 s2k) satisfying     N'/s'1 = N''/s''1     we obtain an OA(N = N'+N'', (s'1+s''1)1 s2k).
For example, from an OA(16, 41 212) and an OA(28, 71 212) we obtain an OA(44, 111 212).
2. Combining an OA(N', s11 s2k) and an OA(N'', s11 s2k) we obtain an OA(N = N'+N'', s11 s2k).
For example, from an OA(24, 61 214) and an OA(36, 61 214) we obtain an OA(60, 61 214).

• Note (t): Trivial construction. Any OA with <= 3 factors is trivial! (Cf. Problem 9.12 in the book.)