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.

翻译：我们为具有非分歧赋值点的线性化代数几何码设计了一种多项式时间解码算法，这类码是函数域上除代数的一类和秩度量赋值码。通过为这些代数建立Serre对偶性和Riemann-Roch定理，我们证明了此类线性化代数几何码的对偶码（我们称之为线性化微分码）在伴随代数上与线性化代数几何码自身重合，并且我们的解码算法是正确的。