Necessary conditions for the existence of divisible designs

Abstract

BOSE and CONNOR [2] proved that a symmetric regular divisible design with w classes of sizes g and joining numbers λ1 and λ2 must satisfy for every prime p the arithmetic condition (d1, (−1)sw)p(d2,(−l)tgw)p=1, where d1=k2−vλ2, d2= k−λ1 s=(w-1)(w-2)/2, t=(v-w)(v-w-1)/2 and (*,*) is the Hilbert symbol. We show that if in addition λ1 ≠ λ2 and the design is fully symmetric divisible then (d1, (−1)s w)p=(d2, (−1)tgw)=1. Our assumption is by a result of CONNOR [5] fulfilled, if d1 and λ1−λ2 are relatively prime. Thus, we can exclude parameters not accessible to the Bose-Connor-Theorem. Our result can be derived from a theorem of RAGHAVARAO [9], and we give the precise assumptions of this theorem. We also discuss arithmetic restrictions for divisible designs which satisfy diverse other rules for the intersection numbers and generalize a result of DEMBOWSKI [6; 2.1.11].