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<title><string language="fre"><![CDATA[1.7. Reed-Solomon Codes]]></string></title>
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<string language="fre"><![CDATA[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.]]></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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