Algebraic-Geometric Codes

• Introduction
• Creation of an Algebraic Geometric Code
  • AlgebraicGeometricCode(S, D): [PlcCrvElt], DivCrvElt → Code
  • AGCode(S, D): [PlcCrvElt], DivCrvElt → Code
  • AlgebraicGeometricDualCode(S, D): [PlcCrvElt], DivCrvElt → Code
  • AGDualCode(S, D): [PlcCrvElt], DivCrvElt → Code
  • HermitianCode(q, r): RngIntElt, RngIntElt → Code
  • Example: Algebraic Geometric Code from a Genus 1 Curve
  • Example: Algebraic Geometric Code from a Genus 4 Curve
• Properties of AG–Codes
  • IsWeaklyAG(C): Code → BoolElt
  • IsWeaklyAGDual(C): Code → BoolElt
  • IsAlgebraicGeometric(C): Code → BoolElt
  • IsStronglyAG(C): Code → BoolElt
• Access Functions
  • Curve(C): Code → Crv
  • GeometricSupport(C): Code → DivCrvElt
  • Divisor(C): Code → DivCrvElt
  • GoppaDesignedDistance(C): Code → RngIntElt
• Decoding AG Codes
  • AGDecode(C, v, Fd): CodeLinFld, ModTupFldElt, DivCrvElt → ModTupFldElt
  • Example: Decoding an Algebraic Geometric Code
• Toric Codes
  • ToricCode(P, q): TorPol, RngIntElt → Code
  • ToricCode(S, q): SeqEnum[TorLatElt], RngIntElt → Code
  • ToricCode(S, q): SetEnum[TorLatElt], RngIntElt → Code
  • Example: Toric Code from a Polygon
• Bibliography