Goppa Codes - Department of Mathematics

Goppa Codes Key One Chung
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1.3 Reed-Solomon Codes
Definition. Lr := {f ∈ Fq[x]|deg(f) ≤ r} � {0}
Note that Lr is a vector subspace over Fq.
Definition. F ∗ q = {α1, . . . , αq−1} and 1 ≤ k ≤ q − 1. Then the
Reed-Solomon code RS(k, q) is defined to be
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RS(k, q) := {(f(α1), . . . , f(αq−1))|f ∈ Lk−1}.
It is an image of a linear transformation ɛ : Lk−1 → F q−1
q
ɛ(f) = (f(α1), . . . , f(αq−1)).
given by
Parameters of RS(k, q) are n = q − 1, dim C = k, and d = n − k + 1.
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