Speaker
Description
Algebraic geometry codes have gained significant attention due to their strong structural properties. In this study, we investigate the construction of linear codes from cubic surfaces in the projective space $PG(3, 13)$. A smooth cubic surface over a finite field is known to contain exactly 27 lines. The configuration of these lines and their intersection points, specifically the Eckardt points, determines the isomorphism class of the surface.We focus on the classification of these surfaces over the finite field $GF(13)$. Using the computational algebra software Orbiter, we analyze surfaces with varying numbers of Eckardt points (specifically classes with 4, 6, 10, and 18 points). We employ the Clebsch map to relate the geometry of 6 points in the projective plane $P^2$ to the cubic surface in $P^3$.The main objective of this work is to construct linear codes by utilizing the incidence matrices of lines and points on these surfaces. We explore the potential of these geometric structures to generate Locally Recoverable Codes (LRC) and analyze their parameters $[n, k, d]$. Furthermore, we investigate whether these constructions yield Maximum Distance Separable (MDS) codes by checking the Singleton bound $d \le n - k + 1$. This research aims to bridge classical algebraic geometry with modern coding theory applications.
Affiliation
Gebze Technical University
