Ressource pédagogique : 1.7. Reed-Solomon Codes

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Reed-Solomon codes were introduced by Reed and Solomon in the 1960s. These codes are still used in storage device, from compact-disc player to deep-space application. And they are widely used mainly because of two features: first of all, because they are MDS code, that is, they attain the maximum error detection and correction capacity. The second thing is that they have efficient decoding algorithms. Reed-Solomon codes are particularly useful for burst error correction, that is, they are effective for channels that have memory.So, suppose that we consider n and k nonnegative integers such that they verify this inequality. Now, we take an n-tuple a of elements from the field that are all different. And we take an n-tuple b of elements from the field which are non-zero.The polynomial vector space of all polynomials that have degree at most k will be denoted by Lk. This is a vector space and the polynomial addition and scalar multiplication are defined in the obvious manner. One basis for this vector space is the monomial basis, this one. Now, we consider the evaluation map at the elements a and b. So, the evaluation map of a polynomial f arise from evaluating the polynomial f  at a and scaling by b. The Generalized Reed-Solomon codes of dimension k associated to the pair a, b is defined as the image of the vector space L and this evaluation map. So, this is the definition of Generalized Reed-Solomon codes. The element a will be denoted as code locators and the element b will be defined as the column multipliers.

"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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