The research on orthogonal Steiner systems S$(t,k,v)$ was initiated in 1968. For $(t,k)\in \{(2,3),(3,4)\}$, this corresponds to orthogonal Steiner triple systems (STSs) and Steiner quadruple systems (SQSs), respectively. The existence problem of a pair of orthogonal STSs or SQSs was settled completely thirty years ago. However, for Steiner systems with $t\ge 3$ and $k\ge 5$, only two small examples of orthogonal pairs were known to exist before this work. An infinite family of orthogonal Steiner systems S$(3,5,v)$ is constructed in this paper. In particular, the existence of a pair of orthogonal Steiner systems $\mathrm{S}(3,5,4^m+1)$ is established for any even $m\ge 2$; additionally a pair of orthogonal G-designs G$(\frac{4^m+1}{5},5,5,3)$ is displayed for any odd $m\ge 3$. The construction is based on the Steiner systems admitting 3-transitive automorphism groups supported by elementary symmetric polynomials. Moreover, 50 mutually orthogonal Steiner systems S$(5,8,24)$ are shown to exist.

正交Steiner系统S的研究$（t,k,v）$于 1968 年发起。$（t,k）\epsilon \{（2,3）,（3,4）\}$，这分别对应于正交斯坦纳三重系统（STS）和斯坦纳四重系统（SQS）。一对正交STS或SQS的存在问题在三十年前就已得到彻底解决。然而，对于 Steiner 系统$t\ge 3$和$k\ge 5$，在这项工作之前，只知道存在两个正交对的小例子。正交 Steiner 系统 S 的无限族$（3,5,v）$本文构建。特别是，存在一对正交 Steiner 系统$\mathrm{S}（3,5,4^{米}+1）$对于任意偶数成立$米\ge 2$; 另外还有一对正交的 G 设计 G$（\frac{4^{米}+1}{5},5,5,3）$显示任何奇数$米\ge 3$。该构造基于承认基本对称多项式支持的 3 传递自同构群的 Steiner 系统。此外，50个相互正交的斯坦纳系统S$（5,8,24）$都显示存在。