Geometrical association schemes and fractional factorial designs

In this paper an attempt is made to throw light on the algebraic structure of symmetrical s^-fractional factorial designs, where s is not necessary 2 but a prime power. For such purpose a geometrical factorial association scheme of PG(& — 1, s)-type and the corresponding s*"^-fractional factorial association scheme are introduced in sections 2 and 3 respectively. The corresponding association algebras Wί(PG(k — 1, s)) and $l(s~ — Fr) are also introduced there. Mutually orthogonal idempotents of those algebras are given in section 4. The notion of fractionally similar mapping is introduced in section 5 and the relationship between 2ί(PG(& — 1, s)) and %(s~—Fr) is investigated there. A general definition of the classical notion of aliases is given in section 6. Blocking of the fractional factorial designs is discussed in section 7 in relation to the notion of partial confounding and the pseudo-block factors. The following notation is used throughout this paper: