Intersections of Algebraic Geometry and Coding Theory: A Study of Error-Correcting Codes

Algebraic Geometry and Coding Theory, two fields initially perceived as distinct, have intersected in profound ways, particularly in error-correcting codes.This study examines the complex interactions that exist between these two fields, with an emphasis on creating and exploiting algebraic-geometric codes (AG codes).Regarding efficiency and error correction, AG codes provide plenty of advantages over conventional linear codes by utilizing the characteristics of algebraic curves and surfaces.This study delves into the mathematical foundations of these codes, their construction, and their performance in various communication scenarios.