On the Block Structure of Certain PBIB Designs with Two Associate Classes Having Triangular and $L_2$ Association Schemes

Abstract

The PBIB designs [2] with two associate classes are classified in [3] as 1. Group Divisible, 2. Simple, 3. Triangular, 4. Latin Square type with $i$ constraints, and 5. Cyclic. Group Divisible designs are divided into three types [1]: 1. Singular, 2. Semi-regular, and 3. Regular. It has been proved [1] that every block of a Semi-regular Group Divisible design contains $k/m$ treatments from each of the $m$ groups of the association scheme. In this note we prove analogous results in the case of certain PBIB designs with triangular and $L_2$ association schemes.