The Point-Hyperplane Geometry - Relatively Universal Embedding and Associated Code

Explore the mathematical foundations of point-hyperplane geometry through this conference talk that examines the relatively universal embedding properties and associated coding theory applications. Delve into the geometric structure $\Gamma$ defined on an $(n+1)$-dimensional vector space over an arbitrary field $\mathbb{K}$, where points are pairs $(p, H)$ of points and hyperplanes in projective space $\mathrm{PG}(V)$ with the incidence relation $p \in H$. Learn about the Segre embedding $\varepsilon: \Gamma \rightarrow \mathrm{PG}(M_0)$ that maps geometry points to traceless matrices, creating the image $\Lambda_1$ represented by pure tensors $x \otimes \xi$ satisfying $\xi(x)=0$. Discover how field automorphisms enable twisted versions $\varepsilon_\sigma$ of the embedding, leading to modified geometric structures $\Lambda_\sigma$ with points represented by tensors $x^\sigma \otimes \xi$. Examine the universality problem for the Segre embedding, understanding how the answer depends critically on the underlying field properties and extends previous results for the case $n=2$. Investigate the coding theory applications when $\mathbb{K}=\mathbb{F}_q$ is a finite field, analyzing the linear codes $\mathcal{C}(\Lambda_1)$ and $\mathcal{C}(\Lambda_\sigma)$ arising from these projective systems. Study the determination of code parameters, weight distributions, and geometric characterizations of minimum and maximal weight codewords, connecting abstract algebraic geometry with practical coding theory applications.