  We present an alternative method for computing primary decomposition of
zero-dimensional ideals over finite fields. Based upon the further
decomposition of the invariant subspace of the Frobenius map acting on the
quotient algebra in the algorithm given by S. Gao, D. Wan and M. Wang in 2008,
we get an alternative approach to compute all the primary components at once.
As one example of our method, an improvement of Berlekamp's algorithm by
theoretical considerations which computes the factorization of univariate
polynomials over finite fields is also obtained.
