Binary Goppa code: Difference between revisions

Revision as of 00:36, 7 December 2012

Revision as of 17:12, 7 December 2012

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<math>v(x)</math> is reduced to polynomials <math>a(x)</math> and <math>b(x)</math> using the [[extended euclidean algorithm]], so that <math>a(x)=b(x)\cdot v(x) \mod g(x)</math>, while <math>\deg(a)\leq\lfloor t/2 \rfloor</math> and <math>\deg(b)\leq\lfloor (t-1)/2 \rfloor</math>.

Finally, the ''error correctionlocator polynomial'' is computed as <math>\sigma(x) = a(x)^2 + x\cdot b(x)^2</math>. Note that in binary case, locating the errors is sufficient to correct them, as there's only one other value. Error correction polynomials have to be computed in all non-binary cases as well.

If the original codeword was decodable and the <math>e=(e_0,e_1,\dots,e_{n-1})</math> was the error vector, then