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.

中文翻译：

---

阶 $$2^{2n+1}$$ 的双可分解 Steiner 四重系统

t- $$(v,k,\lambda )$$ 设计是一对 $$(X,\mathcal{B})$$ ，其中 X 是一个 v 元素集，而 $$\mathcal{B}$ $ 是 X 的一组 k 子集，称为块，其特性是 X 的每个 t 子集都包含在 $$\lambda $$ 块中。t- $$(v,k,\lambda )$$ 设计 $$(X,\mathcal{B})$$ 被称为 $$(s,\mu )$$ -resolvable if $$\mathcal {B}$$ 可以划分为 $$\mathcal{B}_1|\cdots |\mathcal{B}_c$$ 使得每个 $$(X,\mathcal{B}_i)$$ 是一个 s- $$(v,k,\mu )$$ 进一步设计，如果每个 $$(X,\mathcal{B}_i)$$ 也是 $$(r,\nu )$$ -resolvable，那么这样的$$(s,\mu )$$ -resolvable t-design 被称为 $$(s,\mu )(r,\nu )$$ -双重可解析。1980 年，Hartman 为 $$v\in \{20,32,44,68,80,104\}$$ 构造了一个 (2, 3)(1, 1)-双可解析 3-(v, 4, 1) 设计和 (2, 3)-resolvable 3- $$(2^7,4,1)$$ 设计。在本文中，我们构造 (2, 3)(1,