Defining Sets of Projective Planes and Biplanes and Their Residuals

Abstract

Suppose $S$ is a defining set of a symmetric $2$-( $v,k,\lambda$ ) design $D$, where $\lambda =1$ or $2$; that is, $D$ is a projective plane or a biplane.In this paper, conditions under which the residual of $S$ is a defining set of the residual of $D$ are investigated.As a consequence, inequalities relating the sizes of smallest defining sets of $D$ and of the residual of $D$ are obtained.The exact sizes of smallest defining sets of $PG(2,5)$, $AG(2,5)$, and the three non-isomorphic $2$-( $10,4,2$ ) designs are determined.