Right Angular Designs

Abstract

Known PBIB designs with two associate classes were classified according to their association schemes by Bose and Shimamoto [3]. Association schemes for three or more associate classes have been little developed except the rectangular association scheme of Vartak [15] and GD $m$-associate designs of Raghavaroa [8] and Hierarchical group divisible designs with $m$-associate classes of Roy [11]. In this paper we define a new association scheme known as right angular scheme which defines four associate class designs. For convenience we call PBIB designs with right angular scheme, right angular designs. In Section 2 we define the association scheme. Sections 4 and 5 deal with the analysis and construction of right angular designs. In Section 6 we study some combinatorial properties and Sections 3 and 7 give the non-existence of some right angular designs.