Binary Goppa code: Difference between revisions

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Fixed indices to match the "Construction and properties" section
Decoding: Fixed a mistake
 
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Codewords belong to the kernel of the syndrome function, forming a subspace of <math>\{0,1\}^n</math>:
 
: <math>\Gamma(g,L)=\left\{ c \in \{0,1\}^n \left|,\Bigg\vert\, \sum_{i=1}^{n} \frac{c_i}{x-L_i} \equiv 0 \modbmod g(x) \right. \right\}</math>
 
The code defined by a tuple <math>(g,L)</math> has dimension at least <math>n-mt</math> and
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If the original codeword was decodable and the <math>e=(e_1,\dots,e_n)</math> was the binary error vector, then
 
: <math>\sigma(x) = \prod_{i=1}^n e_i(x-L_i)^{e_i} </math>
 
Factoring or evaluating all roots of <math>\sigma(x)</math> therefore gives enough information to recover the error vector and fix the errors.