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C.8.4 Fitzgerald-Lax methodAffine codesLet 907#907 be an ideal. Define
908#908
So 909#909 is a zero-dimensional ideal. Define also
910#910.
Every 798#798-ary linear code 78#78 with parameters 830#830 can be seen as an
affine variety code 911#911, that is, the image of a vector space 912#912 of the evaluation map
913#913
914#914
where
915#915, 912#912 is a vector subspace of 53#53 and 916#916 the coset of 265#265 in
917#917 modulo 909#909.Decoding affine variety codesGiven a 798#798-ary 830#830 code 78#78 with a generator matrix 918#918:
In this way we obtain that the code 78#78 is the image of the evaluation above, thus 928#928. 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 929#929 for every error position. We also assume that field equations on 930#930's are included among the polynomials above. Let 78#78 be a 798#798-ary 830#830 linear code such that its dual is written as an affine variety code of the form 931#931. Let 832#832 as usual and 867#867. Then the syndromes are computed by 932#932. Consider the ring 933#933, where 934#934 correspond to the 57#57-th error position and 935#935 to the 57#57-th error value. Consider the ideal 936#936 generated by
937#937
938#938
939#939
For an example see |
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