| ### C.8.4 Fitzgerald-Lax method #### Affine codes Let 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 codes Given a 800#800-ary 832#832 code 78#78 with a generator matrix 920#920: choose 178#178, such that 921#921, and construct 178#178 distinct points 922#922 in 923#923. Construct a Gröbner basis 924#924 for an ideal 253#253 of polynomials from 925#925 that vanish at the points 922#922. Define 926#926 such that 927#927. Then 928#928 span the space 914#914, so that 929#929. 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 `Theorem:` Let 190#190 be the reduced Gröbner basis for 938#938 with respect to an elimination order 942#942. Then we may solve for the error locations and values by applying elimination theory to the polynomials in 190#190. For an example see `sysFL` in decodegb_lib. More on this method can be found in [FL1998].

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