Duality and decoding of linearized Algebraic Geometry codes
This work addresses decoding challenges in coding theory for applications like secure communications, but it is incremental as it builds on existing algebraic geometry code frameworks.
The paper tackles the problem of decoding linearized Algebraic Geometry codes by designing a polynomial-time algorithm and proving its correctness through duality theorems, achieving a decoding solution for a specific family of sum-rank metric codes.
We design a polynomial time decoding algorithm for linearized Algebraic Geometry codes with unramified evaluation places, a family of sum-rank metric evaluation codes on division algebras over function fields. By establishing a Serre duality and a Riemann-Roch theorem for these algebras, we prove that the dual codes of such linearized Algebraic Geometry codes, that we term linearized Differential codes, coincide with the linearized Algebraic Geometry codes themselves over the adjoint algebra, and that our decoding algorithm is correct.