Notice
1.5. Error Correcting Capacity
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Descriptif
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.
Intervention
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