Doubly resolvable Steiner quadruple systems of orders $2^{2n+1}$

Abstract

A t-$(v,k,\lambda )$ design is a pair $(X,\mathcal{B})$, where X is a v-element set and $\mathcal{B}$ is a set of k-subsets of X, called blocks, with the property that every t-subset of X is contained in exactly $\lambda$ blocks. A t-$(v,k,\lambda )$ design $(X,\mathcal{B})$ is said to be $(s,\mu )$-resolvable if $\mathcal{B}$ can be partitioned into $\mathcal{B}_1|\cdots |\mathcal{B}_c$ such that each $(X,\mathcal{B}_i)$ is an s-$(v,k,\mu )$ design, further, if each $(X,\mathcal{B}_i)$ is also $(r,\nu )$-resolvable, then such an $(s,\mu )$-resolvable t-design is called $(s,\mu )(r,\nu )$-doubly resolvable. In 1980, Hartman constructed a (2, 3)(1, 1)-doubly resolvable 3-(v, 4, 1) design for $v\in \{20,32,44,68,80,104\}$ and a (2, 3)-resolvable 3-$(2^7,4,1)$ design. In this paper, we construct (2, 3)(1, 1)-doubly resolvable 3-$(2^{2n+1},4,1)$ designs for all positive integers n.