| 
 | C.8.2 Cooper philosophyComputing syndromes in cyclic code caseLet be an ![$[n,k]$](sing_793.png) cyclic code over  ;  is a splitting field with  being a primitive n-th root of unity. Let  be the complete defining set of  . Let  be a received word with  and  an error vector.
  Denote the corresponding polynomials in ![$F_q[x]/\langle x^n-1 \rangle$](sing_798.png) by  ,  and  , resp. Compute syndromes   where  is the number of errors,  are the error positions and  are the   error values.  Define  and  . Then  are the error locations and  are the error values and
  the syndromes above become generalized power sum functions  CRHT-idealReplace the concrete values above by variables and add some natural restrictions. Introduce
 
  We obtain the following set of polynomials in the variables 
 ![\begin{displaymath}F_C=\{f_j,\epsilon_j,\eta_i,\lambda_i:1\le j\le n-k,1\le i\le e\}\subset F_q[X,Z,Y].\end{displaymath}](sing_819.png)  The zero-dimensional ideal  generated by  is the CRHT-syndrome ideal
  associated to the code  , and the variety  defined by  is the CRHT-syndrome variety,
  after Chen, Reed, Helleseth and Truong. General error-locator polynomialAdding some more polynomials to , thus obtaining some  , it is possible to prove the following Theorem: 
  Every cyclic code  
 
  The general error-locator polynomial  actually is an element of the reduced Gröbner basis of 
 
For an example see  Finding the minimum distanceThe method described above can be adapted to find the minimum distance of a code. More concretely, the following holds:
  Let                Then the number of solutions of  is equal to  times the number of codewords of weight  . And for  , either  has no solutions, which is equivalent to  , or  has some solutions, which is equivalent to  . 
For an example see  | 
|   |  |  |  |  |  |  |  |  |  |