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C.8.4 Fitzgerald-Lax methodAffine codes
Let
be an
ideal. Define
So is a zero-dimensional ideal. Define also . Every -ary linear code with parameters can be seen as an affine variety code , that is, the image of a vector space of the evaluation map where , is a vector subspace of and the coset of in modulo . Decoding affine variety codesGiven a -ary code with a generator matrix :
In this way we obtain that the code is the image of the evaluation above, thus . In the same way by considering a parity check matrix instead of a generator matrix we have that the dual code is also an affine variety code. The method of decoding is a generalization of CRHT. One needs to add polynomials for every error position. We also assume that field equations on 's are included among the polynomials above. Let be a -ary linear code such that its dual is written as an affine variety code of the form . Let as usual and . Then the syndromes are computed by .
Consider the ring
, where
correspond to
the -th error position and to the -th error value. Consider the ideal generated by
For an example see |