On Orthogonal Arrays

Complete enumeration of pure‐level and mixed‐level orthogonal arrays

“…All classes for the factor sets 2 3 and 2 4 and any N are given by Seiden and Zemach [21] and Stufken and Tang [24], respectively. The results on N ≤ 9 tabulated in Table I are wellknown.…”

Orthogonal plans of resolution IV and V

“…For N =48, the maximum number of factors to form an orthogonal resolution V design is five (Seiden and Zemach (1966». There are two non-equivalent orthogonal designs, obtained from the following solutions of the K-algorithm: For design d, we take: 3 times the rows like in (a), 0 times the rows like in (b), 3 times the rows like in (c), 0 times the rows like in (d), 3 times the rows like in (e) and 0 times the rows like in (f), Hence, we get:…”

“…For N =80, the maximum number of factors to form an orthogonal resolution V design is six (Seiden and Zemach (1966)). The following solution given by the K-algorithm is constructed: Hence, we get:…”

On two level symmetrical factorial designs

“…C. Bose [1] showed that the maximum number of factors in a symmetrical factorial design, in which each factor operates at s levels, blocks are of size s ~+~ and no main effect or t-factor (t:>l) or lower order interaction is confounded with blocks, is given by the maximum number of distinct points in an n-dimensional projective space PG (n, s) so that no t points among them are linearly dependent. These considerations have later led to the extensive use of fractional factorial designs and the study of their confounding properties has been approached from several closely related points of view, e.g., geometrically by Bose [1] and Kempthorne [12], as a special case of orthogonal arrays (Rao [14], Bush [6], Bose an Bush [2], Selden and Zemach [16]), as a case of saturated designs introduced in the literature by Box and Hunter [4], [5] followed by many research workers including Draper and Mitchell [7], [8] and through the theory of groups (Fisher [9], [10]). …”

“…Selden and Zemach [16] have established in connection with the construction of orthogonal arrays that an increase in the value of n and t(=2v) by one results in the corresponding increase in the number of constraints by exactly one. It follows that if k is the maximum number of points in PG(n, 2), no t linearly dependent, then k-t-1 is the maximum number of distinct points in PG(n+I, 2), no t+l linearly dependent.…”