# Optimal BIBD-extended designs

## TL;DR

This paper explores how extending known optimal binary designs with BIBD blocks affects their optimality, providing boundaries and characterizations for when these extended designs are optimal under A- and D-criteria.

## Contribution

It investigates the optimality of BIBD-extended designs, establishing boundaries and characterizations for their A- and D-optimality in specific cases.

## Key findings

- BIBD-extended designs can become A- and D-optimal after extension.
- Characterization of RGDs leading to optimal extended designs.
- Extension of D-optimality results to A- and D-optimality among BIBD-extended designs.

## Abstract

Balanced incomplete block designs (BIBDs) are a class of designs with v treatments and b blocks of size k that are optimal with regards to a wide range of optimality criteria, but it is not clear which designs to choose for combinations of v, b and k when BIBDs do not exist. In 1992, Cheng showed that for sufficiently large b, the designs which are optimal with respect to commonly used criteria (including the A- and D- criteria) must be found among (M.S)-optimal designs. In particular, this result confirmed the conjecture of John and Mitchell in 1977 on the optimality of regular graph designs (RGDs) in the case of large numbers of blocks. We investigate the effect of extending known optimal binary designs by repeatedly adding the blocks of a BIBD and find boundaries for the number of block so that these BIBD-extended designs are optimal. In particular, we will study the designs for k=2 and b=v-1 and b=v: in these cases the A- and D-optimal designs are not the same but we show that this changes after adding blocks of a BIBD and the same design becomes A- and D-optimal amongst the collection of extended designs. Finally, we characterise those RGDs that give rise to A- and D-optimal extended designs and extend a result on the D-optimality of the a group-divisible design to A- and D-optimality amongst BIBD-extended designs.