Uniformly resolvable three-wise balanced designs with block sizes four and six

Abstract

A $t$-wise balanced design is said to be resolvable if its block set can be partitioned into parts (called resolution classes) such that each part is itself a partition of the point set. It is uniform if all blocks in each resolution class have the same size. In this paper, it is shown that a uniformly resolvable three-wise balanced design of order $v$ with block sizes four and six exists if and only if $v$ is divisible by 4. These uniformly resolvable three-wise balanced designs are also used to construct the infinite classes of resolvable maximal packings (minimal coverings) of triples by quadruples of order $v$ for $v\equiv 0(mod24)$, augmented resolvable Steiner quadruple systems of order $v$ for $v\equiv 26,58,74(mod96)$ and $(1,2)$-resolvable Steiner quadruple systems of order $v$ for $v\equiv 74(mod96)$.