# Singular

### C.8.3 Generalized Newton identities

The error-locator polynomial is defined by

If this product is expanded,

then the coefficients are the elementary symmetric functions in the error locations

#### Generalized Newton identities

The syndromes and the coefficients satisfy the following generalized Newton identities:

#### Decoding up to error-correcting capacity

We have , since . Furthermore

and . Replace the syndromes by variables and obtain the following set of polynomials in the variables and :

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].