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<title><string language="fre"><![CDATA[1.5. Error Correcting Capacity]]></string></title>
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<string language="fre"><![CDATA[This sequence will be about the
error-correcting capacity of a linear code. We describe the way of
considering the space Fq^n as a metric space. This metric
is necessary to justify the principle of decoding
that is returning the nearest codeword to the received vector. The metric principle is
based on the following concept: the Hamming distance
between two vectors is the number of coordinates in which they differ.
The Hamming weight of a vector is
the number of non-zero coordinates. Here we give some examples.
So, the Hamming distance
between these two vectors is 2, since they have two
coordinates in which they differ.
The Hamming distance
between these two strings is 1 because they just differ in
one letter, and the Hamming weight of these vectors
is 2 since it just has two elements which are
non-zero. The Hamming distance is a metric on the vector space Fq^n. This means that these
functions satisfy the usual properties of a distance, that
is non-negativity, symmetry, the Hamming distance is
invariable under permutation, and it verifies the triangle inequality. The proof of these
properties is left as an exercise. A measure for the
error-correcting capability of a linear code is the minimum
distance, that is the least Hamming distance between two
different codewords of a linear code. As we will see later, the
higher the minimum distance, the more errors the code can correct. The reason is that the
minimum distance determines the packing radius of a code, that
is the largest integer s such that the balls of radius s
centered at the codewords are all disjoint.]]></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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<string language="fre"><![CDATA[Droits réservés à l'éditeur et aux auteurs. 
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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