Duality and decoding of linearized Algebraic Geometry 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.

💡 Research Summary

The paper studies a new family of sum‑rank metric codes called linearized Algebraic Geometry (LAG) codes, which are constructed by evaluating sections of Riemann‑Roch spaces on a division algebra D over the function field K of an algebraic curve. The division algebra is taken as a cyclic algebra D = L_r