Ressource pédagogique : 1.5. Error Correcting Capacity

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Description (résumé)

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.

"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)

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