  A multivariate interpolation formula (MVIF) over finite fields is presented
by using the proposed Kronecker delta function. The MVIF can be applied to
yield polynomial relations over the base field among homogeneous symmetric
rational functions. Besides the property that all the coefficients are coming
from the base field, there is also a significant one on the degrees of the
obtained polynomial; namely, the degree of each term satisfies certain
condition. Next, for any cyclic codes the unknown syndrome representation can
also be provided by the proposed MVIF and also has the same properties. By
applying the unknown syndrome representation and the Berlekamp-Massey
algorithm, one-step decoding algorithms can be developed to determine the error
locator polynomials for arbitrary cyclic codes.
