Notice
1.7. Reed-Solomon Codes
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Descriptif
Reed-Solomon codes were introducedby Reed and Solomon in the 1960s. These codes are stillused in storage device, from compact-disc player todeep-space application. And they are widely usedmainly because of two features: first of all, because theyare MDS code, that is, they attain the maximum errordetection and correction capacity. The second thing is that theyhave efficient decoding algorithms. Reed-Solomon codes areparticularly useful for burst error correction,that is, they are effective for channels that have memory.So, suppose that we considern and k nonnegative integers such that theyverify this inequality. Now, we take an n-tuple a of elements from thefield that are all different. And we take an n-tuple b ofelements from the field which are non-zero.The polynomial vector space of all polynomials that have degreeat most k will be denoted by Lk.
This is a vector space andthe polynomial addition and scalar multiplication aredefined in the obvious manner.
One basis for this vector spaceis the monomial basis, this one. Now, we consider theevaluation map at the elements a and b. So, the evaluation map of a polynomial f arise fromevaluating the polynomial f at a and scaling by b. The GeneralizedReed-Solomon codes of dimension k associated to the pair a, bis defined as the image of the vector space Land this evaluation map. So, this is the definition ofGeneralized Reed-Solomon codes.
The element a will bedenoted as code locators and the element b will bedefined as the column multipliers.
Intervention
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