Abstract: The class of nearly balanced incomplete block designs with concurrence range , or NBBD(, is defined. This extends previous notions of ``most balanced'' designs to cover settings where off-diagonal entries of the concurrence matrix must differ by a positive integer . Sufficient conditions are found for optimality of NBBD(2)'s under type-1 criteria, then used to establish - and -optimality in settings where optimal designs were previously unknown. Some NBBD(3)'s are also found to be uniquely - and -optimal. Included is a study of settings where the necessary conditions for balanced incomplete block designs are satisfied, but no balanced design exists.
Key words and phrases: -optimality, concurrence discrepancy, -optimality, nearly balanced incomplete block design, semi-regular graph design.