On balanced incomplete-block designs with repeated blocks

Balanced incomplete-block designs (BIBDs) with repeated blocks are studied and constructed. We continue work initiated by van Lint and Ryser in 1972 and pursued by van Lint in 1973. We concentrate on constructing ( v , b , r , k , λ ) -BIBDs with repeated blocks, especially those with gcd ( b , r , λ ) = 1 and r ≤ 20 . We obtain new bounds for the multiplicity of a block in terms of the parameters of a BIBD, and improvements to these bounds for a resolvable BIBD. This allows us to answer a question of van Lint about the sufficiency of certain conditions for the existence of a BIBD with repeated blocks.