| ### C.8.2 Cooper philosophy #### Computing syndromes in cyclic code case Let be an 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 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-ideal Replace the concrete values above by variables and add some natural restrictions. Introduce ; since ; , since are either -th roots of unity or zero; , since . We obtain the following set of polynomials in the variables , and : 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 polynomial Adding some more polynomials to , thus obtaining some , it is possible to prove the following Theorem: Every cyclic code possesses a general error-locator polynomial from that satisfies the following two properties: with , where is the error-correcting capacity; given a syndrome corresponding to an error of weight and error locations , if we evaluate the for all , then the roots of are exactly and 0 of multiplicity , in other words The general error-locator 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 `sysCRHT` in decodegb_lib. More on Cooper's philosophy and the general error-locator polynomial can be found in [OS2005]. #### Finding the minimum distance The 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 `sysCRHTMindist` in decodegb_lib. More on finding the minimum distance with Groebner bases can be found in [S2007]. See [OS2005], for the definition of the polynomial above.

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