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Multivariate Interpolation Formula over Finite Fields and Its Applications in Coding Theory

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.