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C.8.2 Cooper philosophyComputing syndromes in cyclic code caseLet 78#78 be an 832#832 cyclic code over 799#799; 710#710 is a splitting field with 4#4 being a primitive n-th root of unity. Let 833#833 be the complete defining set of 78#78. Let 834#834 be a received word with 835#835 and 836#836 an error vector. Denote the corresponding polynomials in 837#837 by 838#838, 813#813 and 839#839, resp. Compute syndromes
840#840
where 503#503 is the number of errors, 841#841 are the error positions and
842#842 are the error values. Define 843#843 and 844#844. Then 845#845 are the error locations and 846#846 are the error values and
the syndromes above become generalized power sum functions
847#847CRHT-idealReplace the concrete values above by variables and add some natural restrictions. Introduce
We obtain the following set of polynomials in the variables 855#855, 856#856 and 857#857:
858#858
The zero-dimensional ideal 859#859 generated by 860#860 is the CRHT-syndrome ideal
associated to the code 78#78, and the variety 861#861 defined by 860#860 is the CRHT-syndrome variety,
after Chen, Reed, Helleseth and Truong.General error-locator polynomialAdding some more polynomials to 860#860, thus obtaining some 862#862, it is possible to prove the following Theorem:Every cyclic code 78#78 possesses a general error-locator polynomial 863#863 from 864#864 that satisfies the following two properties:
The general error-locator polynomial actually is an element of the reduced Gröbner basis of 877#877. Having this polynomial, decoding of the cyclic code 78#78 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 78#78 be the binary 805#805 cyclic code with the defining set 878#878. Let 879#879 and let 880#880 denote the system:
881#881
882#882
883#883
884#884
882#882
885#885
886#886
Then the number of solutions of 880#880 is equal to 887#887 times the number of codewords of weight 348#348. And for 888#888, either 880#880
has no solutions, which is equivalent to 889#889, or 880#880 has some solutions, which is equivalent to 890#890.
For an example see |
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