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Explore the emerging theory of evaluation codes in the sum-rank metric through this mathematical lecture that bridges classical coding theory with modern algebraic constructions. Begin with foundational concepts of sum-rank metric codes and univariate Ore polynomials before delving into the development of linearized Reed-Solomon (LRS) codes as sum-rank analogues of classical Reed-Solomon codes. Discover how linearized Reed-Muller (LRM) codes extend this framework using multivariate Ore polynomials, overcoming length limitations while maintaining optimal parameters. Learn about the sophisticated construction of linearized Algebraic Geometry (LAG) codes through the development of Riemann-Roch spaces over Ore polynomial rings with coefficients in function fields of curves. Examine how classical divisor theory and Riemann-Roch spaces on curves provide the foundation for these advanced constructions, with detailed analysis of dimension and minimum distance bounds. Gain insight into cutting-edge research connecting arithmetic geometry, algebraic coding theory, and cryptographic applications, concluding with an overview of ongoing developments in this rapidly evolving field of mathematics.
