On the type-1 optimality of nearly balanced incomplete block designs with small concurrence range

The class of nearly balanced incomplete block designs with concurrence rang e l, or NBBD(l), is defined. This extends previous notions of "most balance d" designs to cover settings where off-diagonal entries of the concurrence matrix must differ by a positive integer l greater than or equal to 2. Suff icient conditions are found for optimality of NBBD(2)'s under type-1 criter ia, then used to establish A- and D-optimality in settings where optimal de signs were previously unknown. Some NBBD(3)'s are also found to be uniquely A- and D-optimal. Included is a study of settings where the necessary cond itions for balanced incomplete block designs are satisfied, but no balanced design exists.