ETD

• algebraic coding theory
• algebraic geometry
• algebraic geometry codes
• Artin-Schreier surfaces
• availability
• curves
• divisors
• elliptic curves
• elliptic surfaces
• evaluation codes
• fibred surfaces
• finite fields
• hierarchical locality
• information theory
• linear codes
• locality
• locally recoverable codes
• minimum distance
• parameters
• recovery sets
• surfaces
• torsion points

This thesis investigates Algebraic Geometry (AG) codes, focusing on Locally Recoverable Codes (LRC), which allow the recovery of corrupted data by accessing only a small portion of the remaining information. After introducing the fundamentals of linear coding theory and its main parameters, the work explores the construction of AG codes from algebraic curves and surfaces over finite fields. The research concentrates on algebraic surfaces, representing the first step beyond curves, and highlights the additional challenges related to divisor theory in higher dimensions.
The study then examines LRCs endowed with availability and hierarchical locality, properties that are particularly relevant for distributed storage systems. The thesis presents three new constructions on elliptic surfaces over finite fields, designed to achieve, respectively, availability, hierarchical locality, and a combination of both. These constructions exploit the group structure of elliptic curves and the fibration of elliptic surfaces to define multiple and nested recovery sets. Finally, the parameters of the constructed codes are estimated, and possible directions for future research are proposed, such as extending the results to other fibred surfaces or exploring the potential of higher-dimensional abelian varieties as sources of LRCs with enhanced properties.