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This course focuses on the study of algebraic geometry codes, also known as Goppa codes. Like Reed-Solomon or Reed-Muller codes, AG codes involve polynomial (or rather rational function) evaluations. However, AG codes differ in that the evaluation points are selected from a specifically chosen algebraic curve. This seemingly minor distinction leads to a substantial increase in the complexity of the mathematics required to construct, understand, and analyze these codes. Indeed, the analysis includes deep results such as the Riemann-Roch Theorem and the Riemann Hypothesis over algebraic function fields, which is, in fact, a famous theorem known as the Hasse-Weil Theorem. Throughout this course, we will develop the intricate and beautiful mathematics that underpin AG codes and provide constructions of codes that surpass the Gilbert-Varshamov bound (for sufficiently large constant size alphabets). The course content is fundamentally algebraic. We will work with rings and groups, focusing

--> 2024 AGC | homepage top of page ​ Algebraic Geometry Codes Fall 2024/5 About the course ​ This course focuses on the study of algebraic geometry codes, also known as Goppa codes. Like Reed-Solomon or Reed-Muller codes, AG codes involve polynomial (or rather rational function) evaluations. However, AG codes differ in that the evaluation points are selected from a specifically chosen algebraic curve . This seemingly minor distinction leads to a substantial increase in the complexity of the mathematics required to construct, understand, and analyze these codes. Indeed, the analysis includes d

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