Duality and decoding of linearized Algebraic Geometry codes

TL;DR

This paper introduces a polynomial-time decoding algorithm for a new class of algebraic geometry codes based on division algebras, utilizing duality and Riemann-Roch theorems to establish their properties.

Contribution

It develops a novel decoding algorithm for linearized Algebraic Geometry codes and proves their duality with linearized Differential codes using algebraic geometry tools.

Findings

Decoding algorithm operates in polynomial time.

Dual codes are shown to be linearized Differential codes.

Theoretical foundations include Serre duality and Riemann-Roch theorem.

Abstract

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.