Resistance of Balanced Incomplete Block Designs

Abstract

Hedayat and John (1974) studied the case of resistance of a balanced incomplete block design (BIBD) to the removal of one treatment. They showed that a necessary and sufficient condition for a BIBD to be resistant (that is, retain its variance balance) upon removal of a single treatment is that the two proper subdesigns created by this removal be BIBDs. Herein their results on the structure of resistant BIBDs are generalized and the relationship of such designs to $t$-designs (which are of interest in combinatorial theory) is shown. The main result is a structure theorem which states that a BIBD in $v$ treatments and $k$ plots per block is resistant to the removal of any subset of a specific set of $n$ treatments, $n < k < v$, if and only if all of the $2_n$ proper subdesigns formed when all $n$ treatments are removed are BIBDs in all $v - n$ remaining treatments. This leads to a lower bound (of $2^n(v - n))$ on the number of blocks required for fully locally resistant BIBDs which are not essentially trivial. Examples of resistant designs are given.