Let $V$ be an $(n+1)$-dimensional vector space over an arbitrary field $\mathbb{K}$ and denote by $\mathrm{PG}(V)$ the corresponding projective space. Define $\Gamma$ as the point-hyperplane geometry of $\mathrm{PG}(V)$, whose points are the pairs $(p, H)$, where $p$ is a point, $H$ is a hyperplane of $\mathrm{PG}(V)$ and $p \in H$ and whose lines are the sets $\ell_{p, *}:=\{(p, U): p \in U\}$ or $\ell_{*, H}=\{(x, H): x \in H\}$. The geometry $\Gamma$ is also known as the long root geometry for the special linear group $\mathrm{SL}(n+1, \mathbb{K})$ and admits an embedding (the Segre embedding of $\Gamma$ ) in the projective space $\mathrm{PG}\left(M_0\right)$, where $M_0$ is the vector space of the traceless square matrices of order $n+1$ with entries in the field $\mathbb{K}$. Since $M_0$ is isomorphic to a hyperplane of the vector space $V \otimes V^*$, we explicitly have

$$
\varepsilon: \Gamma \rightarrow \mathrm{PG}\left(M_0\right), \quad \varepsilon((\langle x\rangle,\langle\xi\rangle))=\langle x \otimes \xi\rangle,
$$

with $x \in V \backslash\{0\}, \xi \in V^* \backslash\{0\}$. The image $\Lambda_1:=\varepsilon(\Gamma)$ of $\varepsilon$ is represented by the pure tensors $x \otimes \xi$ with $x \in V$ and $\xi \in V^*$ such that $\xi(x)=0$.

If the underlying field $\mathbb{K}$ admits non-trivial automorphisms, for $1 \neq \sigma \in \operatorname{Aut}(\mathrm{K})$, then it is possible to define a 'twisted version' $\varepsilon_\sigma$ of $\varepsilon$ as follows

$$
\varepsilon_\sigma: \Gamma \rightarrow \mathrm{PG}\left(V \otimes V^*\right), \varepsilon_\sigma((\langle x\rangle,\langle\xi\rangle))=\left\langle x^\sigma \otimes \xi\right\rangle,
$$

where $x^\sigma:=\left(x_i{ }^\sigma\right)_{i=1}^{n+1}$.
Consequently, the points of $\Lambda_\sigma:=\varepsilon_\sigma(\Gamma)$ are represented by pure tensors of the form $x^\sigma \otimes \xi$, under the condition $\xi(x)=0$.

In the first part of the talk I will address the problem of the universality of the Segre embedding $\varepsilon$ for $\Gamma$ proving that the answer to this question depends on the underlying field $\mathbb{K}$ and generalizing a previous result for $n=2$ (see recent work of I. Cardinali, L. Giuzzi, A. Pasini).

In the second part of the talk, I shall focus on the case where $\mathbb{K}=\mathbb{F}_q$ is a finite field of order $q$. Thus, regarding $\Lambda_1$ and $\Lambda_\sigma$ as projective systems of $\mathrm{PG}\left(M_0\right)$ respectively $\mathrm{PG}\left(V \otimes V^*\right)$, I will consider the linear codes $\mathcal{C}\left(\Lambda_1\right)$ and $\mathcal{C}\left(\Lambda_\sigma\right)$ arising from them. I shall determine the parameters of $\mathcal{C}(\Lambda)$ and $\mathcal{C}\left(\Lambda_\sigma\right)$ as well as their weight list. I will also give a (geometrical) characterization of some of the words of these codes having minimum or maximal weight (see recent work of I. Cardinali, L. Giuzzi).[-]

In general the computation of the weight enumerator of a code is hard and even harder so for the coset leader weight enumerator. Generalized Reed Solomon codes are MDS, so their weight enumerators are known and its formulas depend only on the length and the dimension of the code. The coset leader weight enumerator of an MDS code depends on the geometry of the associated projective system of points. We consider the coset leader weight enumerator of $F_{q}$-ary Generalized Reed Solomon codes of length q + 1 of small dimensions, so its associated projective system is a normal rational curve. For instance in case of the $\left [ q+1,3,q-1 \right ]_{q}$ code where the associated projective system of points consists of the q + 1 points of a plane conic, the answer depends whether the characteristic is odd or even. If the associated projective system of points of a $\left [ q+1,4,q-2 \right ]_{q}$ code consists of the q + 1 points of a twisted cubic, the answer depends on the value of the characteristic modulo 6.[-]

A linear error correcting code is a subspace of a finite-dimensional space over a finite field with a fixed coordinate system. Such a code is said to be locally recoverable with locality r if, for every coordinate, its value at a codeword can be deduced from the value of (certain) r other coordinates of the codeword. These codes have found many recent applications, e.g., to distributed cloud storage.
We will discuss the problem of constructing good locally recoverable codes and present some constructions using algebraic surfaces that improve previous constructions and sometimes provide codes that are optimal in a precise sense.[-]