Nontrivial $t$-designs in polar spaces exist for all $t$

article Nontrivial $t$-designs in polar spaces exist for all $t$ published Charlene Weiß author 70420 100 department A finite classical polar space of rank $n$ consists of the totally isotropic subspaces of a finite vector space over $\mathbb{F}_q$ equipped with a nondegenerate form such that $n$ is the maximal dimension of such a subspace. A $t$-$(n,k,\lambda)$ design in a finite classical polar space of rank $n$ is a collection $Y$ of totally isotropic $k$-spaces such that each totally isotropic $t$-space is contained in exactly $\lambda$ members of $Y$. Nontrivial examples are currently only known for $t\leq 2$. We show that $t$-$(n,k,\lambda)$ designs in polar spaces exist for all $t$ and $q$ provided that $k>\frac{21}{2}t$ and $n$ is sufficiently large enough. The proof is based on a probabilistic method by Kuperberg, Lovett, and Peled, and it is thus nonconstructive. 2025 eng Des. Codes Cryptogr.10.1007/s10623-024-01471-1 93971 - 981 @article{Weiß_2025, title={Nontrivial $t$-designs in polar spaces exist for all $t$}, volume={93}, DOI={<a href="https://doi.org/10.1007/s10623-024-01471-1">10.1007/s10623-024-01471-1</a>}, journal={Des. Codes Cryptogr.}, author={Weiß, Charlene}, year={2025}, pages={971–981} } Weiß C. Nontrivial $t$-designs in polar spaces exist for all $t$. Des Codes Cryptogr. 2025;93:971-981. doi:<a href="https://doi.org/10.1007/s10623-024-01471-1">10.1007/s10623-024-01471-1</a> Weiß, Charlene. “Nontrivial $t$-Designs in Polar Spaces Exist for All $t$.” Des. Codes Cryptogr., vol. 93, 2025, pp. 971–81, doi:<a href="https://doi.org/10.1007/s10623-024-01471-1">10.1007/s10623-024-01471-1</a>. C. Weiß, Des. Codes Cryptogr. 93 (2025) 971–981. Weiß, Charlene. “Nontrivial $t$-Designs in Polar Spaces Exist for All $t$.” Des. Codes Cryptogr. 93 (2025): 971–81. <a href="https://doi.org/10.1007/s10623-024-01471-1">https://doi.org/10.1007/s10623-024-01471-1</a>. C. Weiß, “Nontrivial $t$-designs in polar spaces exist for all $t$,” Des. Codes Cryptogr., vol. 93, pp. 971–981, 2025, doi: <a href="https://doi.org/10.1007/s10623-024-01471-1">10.1007/s10623-024-01471-1</a>. Weiß, C. (2025). Nontrivial $t$-designs in polar spaces exist for all $t$. Des. Codes Cryptogr., 93, 971–981. <a href="https://doi.org/10.1007/s10623-024-01471-1">https://doi.org/10.1007/s10623-024-01471-1</a> 502992024-01-08T14:39:54Z2026-02-25T13:51:50Z