|  C.8.5 Decoding method based on quadratic equations 
  Preliminary definitions 
  Let 
 be a basis of  and let  be the  matrix with  as rows. The unknown syndrome  of a word  w.r.t  is the column vector  with entries  for  . 
For two vectors 
 define  . Then  is a linear combination of  , so there are constants  such that  The elements  are the structure constants of the basis  . 
Let  be the  matrix with  as rows (  ).
  Then  is an ordered MDS basis and  an MDS matrix if all
  the  submatrices of  have rank  for all  . 
  Expressing known syndromes Let be an  -linear code with parameters ![$[n,k,d]$](sing_766.png) . W.l.o.g  .  is a check matrix of  .
  Let  be the rows of  .
  One can express  with some  .
  In other words  where  is the  matrix with entries  . 
Let 
 be a received word with  and  an error vector.
  The syndromes of  and  w.r.t  are equal and known: 
 They can be expressed in the unknown syndromes of
  w.r.t  : 
 since
  and  . 
  Contructing the system 
  Let  be an MDS matrix with structure constants  .
  Define  in the variables  by 
 The ideal
  in ![$F_q[U_1,\dots,U_n]$](sing_935.png) is generated by 
 The ideal
  in ![$F_q[U_1,\dots,U_n,V_1,\dots,V_t]$](sing_938.png) is generated by 
 Let
  be the ideal in ![$F_q[U_1,\dots,U_n,V_1,\dots,V_t]$](sing_938.png) generated by  and  . 
  Main theorem 
  Let  be an MDS matrix with structure constants  .  Let  be a check matrix of the code  such that  as above.
  Let  be a received word with  the codeword sent
  and  the error vector. Suppose that  and  .
  Let  be the smallest positive integer such that  has a solution  over the algebraic closure of  . Then 
   and the solution is unique and of multiplicity one satisfying  .
  the reduced Gröbner basis  for the ideal  w.r.t any
  monomial ordering is 
 
 
 where
  is the unique solution. 
For an example see sysQEin  decodegb_lib. More on this method can be found in  [BP2008a]. 
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