Combinatorial designs have been studied for nearly 200 years. 50 years ago, Cameron, Delsarte, and Ray-Chaudhury started investigating their q-analogs, also known as subspace designs or designs over finite fields. Designs can be defined analogously in finite classical polar spaces, too. The definition includes the m-regular systems from projective geometry as the special case where the blocks are generators of the polar space. The first nontrivial such designs for \(t > 1\) were found by De Bruyn and Vanhove in 2012, and some more designs appeared recently in the PhD thesis of Lansdown. In this article, we investigate the theory of classical and subspace designs for applicability to designs in polar spaces, explicitly allowing arbitrary block dimensions. In this way, we obtain divisibility conditions on the parameters, derived and residual designs, intersection numbers and an analog of Fisher’s inequality. We classify the parameters of symmetric designs. Furthermore, we conduct a computer search to construct designs of strength \(t=2\), resulting in designs for more than 140 previously unknown parameter sets in various classical polar spaces over \(\mathbb {F}_2\) and \(\mathbb {F}_3\).

组合设计的研究已有近 200 年的历史。 50 年前，Cameron、Delsarte 和 Ray-Chaudhury 开始研究他们的q类似物，也称为子空间设计或有限域设计。设计也可以在有限的经典极空间中进行类似的定义。该定义包括来自射影几何的m正则系统作为特殊情况，其中块是极空间的生成器。 De Bruyn 和 Vanhove 在 2012 年发现了第一个重要的\(t > 1\)设计，最近 Lansdown 的博士论文中出现了更多设计。在本文中，我们研究了经典和子空间设计理论在极空间设计中的适用性，明确允许任意块尺寸。通过这种方式，我们获得了参数的整除条件、导出设计和残差设计、交集数以及费舍尔不等式的模拟。我们对对称设计的参数进行分类。此外，我们进行计算机搜索来构建强度\(t=2\)的设计，从而在\(\mathbb {F}_2\)和\( \mathbb {F}_3\) 。