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<title><string language="fre"><![CDATA[1.8. Goppa Codes]]></string></title>
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<string language="fre"><![CDATA[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.]]></string></description>
<keyword><string language="fre"><![CDATA[algèbre linéaire]]></string></keyword><keyword><string language="fre"><![CDATA[chiffrement à clé publique]]></string></keyword><keyword><string language="fre"><![CDATA[cryptage des données]]></string></keyword><keyword><string language="fre"><![CDATA[cryptographie]]></string></keyword>
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<date><dateTime>2015-05-05</dateTime></date>
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Ces ressources de cours sont, sauf mention contraire, diffusées sous Licence Creative Commons. L’utilisateur doit mentionner le nom de l’auteur, il peut exploiter l’œuvre sauf dans un contexte commercial et il ne peut apporter de modifications à l’œuvre originale.]]></string>
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<string language="fre"><![CDATA[1: Error-Correcting Codes and Cryptography]]></string>
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