Goppa code

In mathematics, an algebraic geometric code (AG-code), otherwise known as a Goppa code, is a general type of linear code constructed by using an algebraic curve ${\displaystyle X}$ over a finite field ${\displaystyle \mathbb {F} _{q}}$. Such codes were introduced by Valerii Denisovich Goppa. In particular cases, they can have interesting extremal properties. They should not be confused with Binary Goppa codes that are used, for instance, in the McEliece cryptosystem.

Traditionally, an AG-code is constructed from a non-singular projective curve X over a finite field ${\displaystyle \mathbb {F} _{q}}$ by using a number of fixed distinct ${\displaystyle \mathbb {F} _{q}}$ -rational points on ${\displaystyle \mathbf {X} }$:

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