BIB(8, 56, 21, 3, 6) and BIB(10, 30, 9, 3, 2) Designs with Repeated Blocks

Abstract If there are less than b distinct blocks in a BIB design with b blocks then we say the design has repeated blocks. The set of distinct blocks of a design is called the support of the design. BIB designs with repeated blocks, besides being optimal, have special applications in the design of experiments and controlled samplings. Construction of BIB(ν, b , r , k , λ) designs with repeated blocks becomes complicated whenever the three parameters b , r , and λ are relatively prime. BIB(8, 56, 21, 3, 6) designs are examples of such designs with the smallest number of varieties . BIB(10, 30, 9, 3, 2) designs are such designs with the smallest number of blocks . We make an interesting observation about BIB(8, 56, 21, 3, 6) designs and give a table of such designs with 30 different support sizes. We prove, by construction, that a BIB(10, 30, 9, 3, 2) design exists if and only if the support size belongs to {21, 23, 24, 25, 26, 27, 28, 29, 30}. Other results are also given.