Goppa Codes - Department of Mathematics

Goppa Codes Key One Chung
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2.2 Algebraic and Projective Curves
k denotes the algebraic closure of the field k.
Definition. We define the affine place A 2 (k) to be the set k 2 .
Definition. If f ∈ k[x, y], then the affine(algebraic) curve is defined to be
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Cf := {P ∈ A 2 |f(P ) = 0}.
Let X be an algebraic curve over k. Then,
I(X) = {f ∈ k[x, y]|f(P ) = 0 for all P ∈ X}
is an ideal of k[x, y]. The quotient ring Γ(X) = k[x, y]/I(X) is called the
coordinate ring of X.
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