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

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 $\mathrm{PG}(V)\otimes \mathrm{PG}(V^{\ast })$ in $\mathrm{PG}(V\otimes V^{\ast })$. Consider the subvariety ${\Lambda }_1$ of $\Lambda$ represented by the pure tensors $x\otimes \xi$ with $x\in V$ and $\xi \in V^{\ast }$ such that $\xi (x)=0$. Regarding ${\Lambda }_1$ as a projective system of $\mathrm{PG}(V\otimes V^{\ast })$, 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.

31 pages/revised version

References in corpus (7)

• A geometric characterization of minimal codes and their asymptotic performance
• Schubert Varieties, Linear Codes and Enumerative Combinatorics
• Automorphism groups of Grassmann codes
• Minimum distance of Symplectic Grassmann codes
• Linear Codes associated to Determinantal Varieties
• Minimum distance of Line Orthogonal Grassmann Codes in even characteristic
• Line Hermitian Grassmann Codes and their Parameters