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Binary Goppa code is defined by a [[polynomial]] <math>g(x)</math> of degree <math>t</math> over a [[finite field]] <math>GF(2^m)</math> without multiple zeros, and a sequence <math>L</math> of <math>n</math> distinct elements from <math>GF(2^m)</math> that aren't roots of the polynomial:
<!-- \mathbb{Z}_n is a ring (or an additive group), so unless you need i,j to be elements of that ring/group,
: <math>\forall i,j \in \mathbb{Z}_n: L_i \in GF(2^m) \and L_i \neq L_j \and g(L_i) \neq 0</math>▼
set notation should be used instead this notation -->
▲: <math>\forall i,j \in \
Codewords belong to the kernel of syndrome function, forming a subspace of <math>\{0,1\}^n</math>:
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