A three-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, q2)

First we define relations between the v = (s2+s+1}(s+l) flags (point-line incident pairs) of a finite projective plane of order s. Two flags a =(p,i) and b =(p' ,l'), where p and p' are two points and l and l' are two lines of the projective plane, are defined to be first associates if either p = p' or l = l'; second associates if p # p', l # 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+l) flags with n l = 2s, n 2 = 2s2 and n 3 = s3 (n i denotes the number of i-th associates of a given flag, i = 1,2,3) and the association matrices are 3 [ 0 P3 = (Pij) = 2 2(s-l} s s(s-l} 2 s s-1 s 2s(s-l} 2 4(s-l} 2 2(s-l} 2(s-l} I 2(s-I}2. (s-I}(s2-s+1) If a finite projective plane of order s admits a subplane of order q, then 2 2 2 it is known (Bruck, 1955) that either s =q or s ~ q +q. If s =q , then the subplane is called a Boer subptane. In a Desarguesian finite projective plane 2 2 of order q • PG(2,q }. all subplanes are Baer subplanes of order q and there-2-3 3 2 2 are b =q (q +1)(q +1) Baer subplanes in PG(2,q). Next, we consider the incidence of the flags and the Baer subplanes (blocks) of PG(2,q2). We show that every flag occurs in r = q3(q+l)2 Baer subplanes; a pair of flags which are first associates occur together in Al = q2(q+l)2 blocks; a pair of flags which are second associates occur together in 2 A 2 = q(q+l) blocks; any two flags which are third associates occur together in 2 2 ~ = (q+l) blocks. Each Baer subplane is incident with k = (q +q+l)(q+l) 2 flags. Thus the incidence of the flags and the Baer subplanes of PG(2,q) defines an incomplete block design (called a partially balanced incomplete 4 2 2 3 3 2 block design) with parameter, v = (q +q +1)(q +1), b = q (q +1)(q +1), r = 3 2 2 2 2 2 2 q (q+l) , k = (q +q+l)(q+l), Al = q (q+l) A 2 = q(q+l) and A …