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C.8.3 Generalized Newton identities

The error-locator polynomial is defined by

891#891
If this product is expanded,
892#892
then the coefficients 893#893 are the elementary symmetric functions in the error locations 845#845
894#894

Generalized Newton identities

The syndromes 895#895 and the coefficients 893#893 satisfy the following generalized Newton identities:

896#896

Decoding up to error-correcting capacity

We have 897#897, since 898#898. Furthermore

899#899
and 900#900. Replace the syndromes by variables and obtain the following set of polynomials 901#901 in the variables 902#902 and 903#903:
904#904
905#905
906#906
907#907
908#908

For an example see sysNewton in decodegb_lib. More on this method and the method based on Waring function can be found in [ABF2002]. See also [ABF2008].


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