Notice
1.8. Goppa Codes
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Descriptif
In this session, we willtalk about another family of codes that have an efficientdecoding algorithm: the Goppa codes. One limitation of thegeneralized Reed-Solomon codes is the fact that the length isbounded by the size of the field over which it is defined. This implies that these codes areuseful when we use a large field size. In the sequence, we'llpresent a method to obtain a new code over small alphabets byexploiting the properties of the generalized Reed-Solomon codes. So, the idea is to construct a generalizedReed-Solomon code over a sufficiently large extensionof a field and extract only those codewords that liecompletely in the field. Let a be a n-tuple ofelements from the field Fq^n which are all differentand let b be an n-tuple of elements from the field Fq^n which are non-zeros. Then, the alternant codesassociated to the pair a,b and parameter r is the Fqlinear restriction of the generalized Reed-Solomon codes ofdimension r associated to the pair a,b. a will be called the supportand b the column multiplier. So, the alternant code associated to the parameters a and b isa linear code with dimension greater than n - mr andminimum distance greater than r + 1 And the proof is very easy. First of all, recall thatthe dual of a generalized Reed-Solomon code is again ageneralized Reed-Solomon code.And this new generalizedReed-Solomon code has parameters n, n - r and r + 1 since the generalized Reed-Solomon code is MDS. Thus, the alternant codeassociated to the pair a and b can be defined by r paritycheck equations over Fq^m and mr parity check equations over Fq. So, the dimension of thealternant code must be at least n - mr. Moreover, theminimum distance of an alternant code is at least theminimum distance of a generalized Reed-Solomon code since thegeneralized Reed-Solomon code is a subset of the alternant code.
Intervention
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