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.