
C.8.2 Cooper philosophyComputing syndromes in cyclic code caseLet be an cyclic code over ; is a splitting field with being a primitive nth 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 by , and , resp. Compute syndromeswhere 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 CRHTidealReplace the concrete values above by variables and add some natural restrictions. Introduce
We obtain the following set of polynomials in the variables
,
and
:
The zerodimensional ideal generated by is the CRHTsyndrome ideal associated to the code , and the variety defined by is the CRHTsyndrome variety, after Chen, Reed, Helleseth and Truong. General errorlocator polynomialAdding some more polynomials to , thus obtaining some , it is possible to prove the following Theorem:Every cyclic code possesses a general errorlocator polynomial from that satisfies the following two properties:
The general errorlocator polynomial actually is an element of the reduced Gröbner basis of . Having this polynomial, decoding of the cyclic code reduces to univariate factorization.
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 be the binary cyclic code with the defining set
. Let and let denote the system:
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 