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A 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:
: <math>\forall i,j \in \{0,\ldots,n-1\}: L_i \in GF(2^m) \land ((i = j) \lor (L_i \neq L_j) )\land g(L_i) \neq 0</math>
Codewords belong to the kernel of syndrome function, forming a subspace of <math>\{0,1\}^n</math>:
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