Vidéo pédagogique
Langue :
Irene Marquez-Corbella (Intervention), Nicolas Sendrier (Intervention), Matthieu Finiasz (Intervention)
Conditions d'utilisation
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.
DOI : 10.60527/mz84-6420
Citer cette ressource :
Irene Marquez-Corbella, Nicolas Sendrier, Matthieu Finiasz. Inria. (2015, 5 mai). 1.5. Error Correcting Capacity , in 1: Error-Correcting Codes and Cryptography. [Vidéo]. Canal-U. (Consultée le 14 juillet 2024)

1.5. Error Correcting Capacity

Réalisation : 5 mai 2015 - Mise en ligne : 20 février 2017
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This sequence will be about theerror-correcting capacity of a linear code. We describe the way ofconsidering the space Fq^n as a metric space. This metricis necessary to justify the principle of decodingthat is returning the nearest codeword to the received vector. The metric principle isbased on the following concept: the Hamming distancebetween two vectors is the number of coordinates in which they differ.The Hamming weight of a vector isthe number of non-zero coordinates. Here we give some examples.So, the Hamming distancebetween these two vectors is 2, since they have twocoordinates in which they differ.The Hamming distancebetween these two strings is 1 because they just differ inone letter, and the Hamming weight of these vectorsis 2 since it just has two elements which arenon-zero. The Hamming distance is a metric on the vector space Fq^n. This means that thesefunctions satisfy the usual properties of a distance, thatis non-negativity, symmetry, the Hamming distance isinvariable under permutation, and it verifies the triangle inequality. The proof of theseproperties is left as an exercise. A measure for theerror-correcting capability of a linear code is the minimumdistance, that is the least Hamming distance between twodifferent codewords of a linear code. As we will see later, thehigher the minimum distance, the more errors the code can correct. The reason is that theminimum distance determines the packing radius of a code, thatis the largest integer s such that the balls of radius scentered at the codewords are all disjoint.


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