|  |  C.8.4 Fitzgerald-Lax method 
  Affine codes 
  Let 
![$I=\langle g_1,\dots,g_m \rangle \subseteq F_q[X_1,\dots,X_s]$](sing_870.png) be an
  ideal. Define 
 So
  is a zero-dimensional ideal. Define also  .
  Every  -ary linear code  with parameters ![$[n,k]$](sing_793.png) can be seen as an
  affine variety code  , that is, the image of a vector space  of the evaluation map 
 
 
 where
 ![$R:=F_q[U_1,\dots,U_s]/I_q$](sing_878.png) ,  is a vector subspace of  and  the coset of  in ![$F_q[U_1,\dots,U_s]$](sing_880.png) modulo  . 
  Decoding affine variety codes 
  Given a  -ary ![$[n,k]$](sing_793.png) code  with a generator matrix  : 
  choose  , such that  , and construct  distinct points  in  .
  Construct a Gröbner basis 
 for an ideal  of polynomials from ![$F_q[X_1,\dots,X_s]$](sing_886.png) that vanish at the points  . Define ![$\xi_i\in F_q[X_1,\dots,X_s]$](sing_887.png) such that  .
  Then 
 span the space  , so that  . 
  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 ![$[n,k]$](sing_793.png) 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 
![$F_q[X_{11},\dots,X_{1s},\ldots,X_{t1},\dots,X_{ts},E_1,\dots,E_t]$](sing_896.png) , where  correspond to
  the  -th error position and  to the  -th error value. Consider the ideal  generated by 
 
 
 
 
 
 
 
Theorem:  Let  be the reduced Gröbner basis for  with respect to an elimination order  .
  Then we may solve for the error locations and values by applying elimination theory to the polynomials in  . 
For an example see sysFLin  decodegb_lib. More on this method can be found in  [FL1998]. 
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