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We provide a theoretical study of Algebraic Geometry codes constructed fromabelian surfaces defined over finite fields. We give a general bound on theirminimum distance and we investigate how this estimation can be sharpened underthe assumption that the abelian surface does not contain low genus curves. Thisapproach naturally leads us to consider Weil restrictions of elliptic curvesand abelian surfaces which do not admit a principal polarization.

Algebraic geometry codes over abelian surfaces containing no absolutely irreducible curves of low genus