Construction of triangular design by an unreduced balanced incomplete block design

Another important and highly investigated category of Partially Balanced Incomplete Block Designs (PBIBDs) with two associate classes is triangular designs. This paper describes a construction procedure of triangular designs with an unreduced Balanced Incomplete Block Design (BIBD). Triangular designs form a special case of PBIBDs, and are therefore also applicable to experimental design, in coding theory, and in finite geometry by their clear associate structure and incidence properties. The suggested construction offers a systematic structure through the parameters and incidence relationships of an unreduced BIBD to generate a triangular scheme of association. The relation between the parameters of the unreduced BIBD and those of the triangular design obtained is determined, and the conditions that are required to be the case are checked. A number of examples are provided to prove the usefulness of the proposed approach. This method underlines a combinatorial association between BIBDs and PBIBDs.