| ### C.8.4 Fitzgerald-Lax method #### Affine codes Let 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 codes Given a 798#798-ary 830#830 code 78#78 with a generator matrix 918#918: choose 177#177, such that 919#919, and construct 177#177 distinct points 920#920 in 921#921. Construct a Gröbner basis 922#922 for an ideal 251#251 of polynomials from 923#923 that vanish at the points 920#920. Define 924#924 such that 925#925. Then 926#926 span the space 912#912, so that 927#927. 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 `Theorem:` Let 189#189 be the reduced Gröbner basis for 936#936 with respect to an elimination order 940#940. Then we may solve for the error locations and values by applying elimination theory to the polynomials in 189#189. For an example see `sysFL` in decodegb_lib. More on this method can be found in [FL1998].

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