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C.8.4 Fitzgerald-Lax methodAffine codesLet 909#909 be an ideal. Define
910#910
So 911#911 is a zero-dimensional ideal. Define also
912#912.
Every 800#800-ary linear code 78#78 with parameters 832#832 can be seen as an
affine variety code 913#913, that is, the image of a vector space 914#914 of the evaluation map
915#915
916#916
where
917#917, 914#914 is a vector subspace of 53#53 and 918#918 the coset of 267#267 in
919#919 modulo 911#911.Decoding affine variety codesGiven a 800#800-ary 832#832 code 78#78 with a generator matrix 920#920:
In this way we obtain that the code 78#78 is the image of the evaluation above, thus 930#930. 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 931#931 for every error position. We also assume that field equations on 932#932's are included among the polynomials above. Let 78#78 be a 800#800-ary 832#832 linear code such that its dual is written as an affine variety code of the form 933#933. Let 834#834 as usual and 869#869. Then the syndromes are computed by 934#934. Consider the ring 935#935, where 936#936 correspond to the 57#57-th error position and 937#937 to the 57#57-th error value. Consider the ideal 938#938 generated by
939#939
940#940
941#941
For an example see |
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