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C.8.2 Cooper philosophyComputing syndromes in cyclic code caseLet 78#78 be an 830#830 cyclic code over 797#797; 708#708 is a splitting field with 4#4 being a primitive n-th root of unity. Let 831#831 be the complete defining set of 78#78. Let 832#832 be a received word with 833#833 and 834#834 an error vector. Denote the corresponding polynomials in 835#835 by 836#836, 811#811 and 837#837, resp. Compute syndromes
838#838
where 501#501 is the number of errors, 839#839 are the error positions and
840#840 are the error values. Define 841#841 and 842#842. Then 843#843 are the error locations and 844#844 are the error values and
the syndromes above become generalized power sum functions
845#845CRHT-idealReplace the concrete values above by variables and add some natural restrictions. Introduce
We obtain the following set of polynomials in the variables 853#853, 854#854 and 855#855:
856#856
The zero-dimensional ideal 857#857 generated by 858#858 is the CRHT-syndrome ideal
associated to the code 78#78, and the variety 859#859 defined by 858#858 is the CRHT-syndrome variety,
after Chen, Reed, Helleseth and Truong.General error-locator polynomialAdding some more polynomials to 858#858, thus obtaining some 860#860, it is possible to prove the following Theorem:Every cyclic code 78#78 possesses a general error-locator polynomial 861#861 from 862#862 that satisfies the following two properties:
The general error-locator polynomial actually is an element of the reduced Gröbner basis of 875#875. 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 803#803 cyclic code with the defining set 876#876. Let 877#877 and let 878#878 denote the system:
879#879
880#880
881#881
882#882
880#880
883#883
884#884
Then the number of solutions of 878#878 is equal to 885#885 times the number of codewords of weight 346#346. And for 886#886, either 878#878
has no solutions, which is equivalent to 887#887, or 878#878 has some solutions, which is equivalent to 888#888.
For an example see |
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