Binary Goppa code: Difference between revisions

Revision as of 17:16, 7 December 2012

Revision as of 18:34, 9 December 2012

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</math>

Note that this form of the parity-check matrix, being composed of a [[Vandermonde matrix]] <math>V</math> and [[diagonal matrix]] <math>D</math>, shares the form with check matrices of [[generalized Reed–Solomonalternant code]]s, thus GRS (and also BCH)alternant decoders can be used on this form. Such decoders usually provide only limited error-correcting capability (in most cases <math>t/2</math>).

For practical purposes, parity-check matrix of a binary Goppa code is usually converted to a more computer-friendly binary form by a trace construction, that converts the <math>t</math>-by-<math>n</math> matrix over <math>GF(2^m)</math> to a <math>mt</math>-by-<math>n</math> binary matrix by writing polynomial cofficients of <math>GF(2^m)</math> elements on <math>m</math> successive rows.