Ressource pédagogique : 1.8. Goppa Codes

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Description (résumé)

In this session, we will talk about another family of codes that have an efficient decoding algorithm: the Goppa codes. One limitation of the generalized Reed-Solomon codes is the fact that the length is bounded by the size of the field over which it is defined. This implies that these codes are useful when we use a large field size. In the sequence, we'll present a method to obtain a new code over small alphabets by exploiting the properties of the generalized Reed-Solomon codes. So, the idea is to construct a generalized Reed-Solomon code over a sufficiently large extension of a field and extract only those codewords that lie completely in the field.  Let a be a n-tuple of elements from the field Fq^n which are all different and let b be an n-tuple of elements from the field Fq^n which are non-zeros. Then, the alternant codes associated to the pair a,b and parameter r is the Fq linear restriction of the generalized Reed-Solomon codes of dimension r associated to the pair a,b. a will be called the support and b the column multiplier. So, the alternant code associated to the parameters a and b is a linear code with dimension greater than n - mr and minimum distance greater than r + 1  And the proof is very easy. First of all, recall that the dual of a generalized Reed-Solomon code is again a generalized Reed-Solomon code. And this new generalized Reed-Solomon code has parameters n, n - r and r + 1 since the generalized Reed-Solomon code is MDS. Thus, the alternant code associated to the pair a and b can be defined by r parity check equations over Fq^m and mr parity check equations over Fq. So, the dimension of the alternant code must be at least n - mr. Moreover, the minimum distance of an alternant code is at least the minimum distance of a generalized Reed-Solomon code since the generalized Reed-Solomon code is a subset of the alternant code.

"Domaine(s)" et indice(s) Dewey

• Analyse numérique (518)
• Théorie de l'information (003.54)
• données dans les systèmes informatiques (005.7)
• cryptographie (652.8)
• Mathématiques (510)

Thème(s)