On the existence and generation of non- trivial BIBDs

Abstract

A balanced incomplete block design (BIBD) is an efficient tool to infer parameters of a linear model. With a BIBD treatments are compared in blocks of size A variety of algebraic methods were developed to determine BIBDs depending on the parameter pair (v,k). Despite the progress in the field, there are many pairs (v,k) without an algebraic solution. For such pairs it is solely known, that a trivial BIBD exists, which is commonly large. In this study we present new results on the existence and generation of BIBDs which are smaller than the trivial ones. Such so-called non- trivial BIBDs exist if a trivial BIBDs is not elementary. With a recursive method of BIBD- generation it is shown that for and there exist non- trivial BIBDs. The same is shown for and Numerical programs were written for the generation of new non- trivial BIBDs, which delivered smaller BIBDs than the recursive method for Commonly used R-programs like “crossdes” and “ibd” failed to determine such BIBDs.