A 3-CLASS ASSOCIATION SCHEME ON THE FLAGS OF A FINITE PROJECTIVE PLANE AND A (PBIB) DESIGN DEFINED BY THE INCIDENCE OF THE FLAGS AND THE BAER SUBPLANES IN PG(2, Q(2))

First we define relations between the v = (s2 + s + 1) (s + 1) flags ( point-line incident pairs) of a finite projective plane of order s. Tw o flags a = (p, l) and b = (p', l'), where p and p' are two points and 1 and l' are two lines of the projective plane, are defined to be fir st associates if either p = p' or l = l': second associates if p not-e qual p', l not-equal l' but either p is incident also with l' or p' is incident also with l: third associates, otherwise. We show that these relations define a three-class association scheme on v = (s2 + s + 1) (s + 1) flags with n1 = 2s, n2 = 2s2 and n3 = s3(n(i) denotes the numb er of i-th associates of a given flag, i = 1, 2, 3) and the associatio n matrices are given in Section 2. If a finite projective plane of ord er s admits a subplane of order q, then it is known (Bruck [2]) that e ither s = q or s greater-than-or-equal-to q + q. If s = q, then the pl ane of order s has a subplane of order q which is called a Baer subpla ne. In a Desarguesian finite projective plane of order q2, PG(2, q), a ll subplanes are Baer subplanes of order q and there are b = q3(q3 + 1 ) (q 2 + 1) Baer subplanes in PG (2, q2). Next, we show that the incid ence of the flags and the Baer subplanes of PG(2, q2) defines an incom plete block design (called a partially balanced incomplete block desig n) with parameter. v = (q4 + q2 + 1)(q2 + 1), b = q 3(q3 + 1) (q 2 +1) , r = q3(q + 1)2, k = (q2 + q + 1)(q + 1), lambda1 = q2(q + 1)2, lambd a2 = q(q + 1)2, lambda3 = (q + 1)2. based on the three-class associati on scheme defined on the flags.