Linear codes arising from the point-hyperplane geometry-Part I: the Segre embedding

TL;DR

This paper investigates a linear code derived from the Segre embedding of a point-hyperplane geometry, determining its parameters, automorphisms, and characterizing key codewords through geometric properties.

Contribution

It introduces a new linear code from the Segre embedding, fully characterizes its parameters, automorphism group, and describes special codewords geometrically.

Findings

The code is minimal.

Parameters and weight distribution are explicitly determined.

Geometric characterization of minimum and second minimum weight codewords.

Abstract

Let $V$ be a vector space over the finite field $\mathbb{F}_q$ with $q$ elements and $\Lambda$ be the image of the Segre geometry $\mathrm{PG}(V)\otimes\mathrm{PG}(V^*)$ in $\mathrm{PG}(V\otimes V^*)$. Consider the subvariety $\Lambda_{1}$ of $\Lambda$ represented by the pure tensors $x\otimes \xi$ with $x\in V$ and $\xi\in V^*$ such that $\xi(x)=0$. Regarding $\Lambda_1$ as a projective system of $\mathrm{PG}(V\otimes V^*)$, we study the linear code $\mathcal{C}(\Lambda_1)$ arising from it. The code $\mathcal{C}(\Lambda_1)$ is minimal code and we determine its basic parameters, itsfull weight list and its linear automorphism group. We also give a geometrical characterization of its minimum and second lowest weight codewords as well as of some of the words of maximum weight.