
C.8.4 FitzgeraldLax methodAffine codes
Let
be an
ideal. Define
So is a zerodimensional 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 