Duality and decoding of linearized Algebraic Geometry codes
🤖 AI Summary
This work investigates linearized algebraic geometry codes constructed from unramified places of division algebras over function fields, focusing on their efficient decoding under the rank metric and their dual structures. It establishes, for the first time, Serre duality and a Riemann–Roch theorem tailored to function fields over division algebras, introduces the notion of linearized differential codes, and proves their equivalence to the original codes under the adjoint algebra. Building on these theoretical foundations, the paper presents the first polynomial-time decoding algorithm that correctly and efficiently decodes such codes. Furthermore, it provides a complete characterization of their duals, thereby laying the groundwork for the theory of noncommutative algebraic geometry codes.
Application Category
📝 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.
Problem
Research questions and friction points this paper is trying to address.
duality
decoding
linearized Algebraic Geometry codes
sum-rank metric
Serre duality
Innovation
Methods, ideas, or system contributions that make the work stand out.
linearized Algebraic Geometry codes
Serre duality
Riemann-Roch theorem
sum-rank metric
polynomial time decoding
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