A note on linearized polynomials and the dimension of their kernels

Recently explicit representations of the class of linearized permutation polynomials and the number of such polynomials were given in Zhou (2008) [4], and Yuan and Zeng (2011) [3], . In this paper, we generalize this result to linearized polynomials with kernel of any given dimension, solving an open problem in Charpin and Kyureghyan (2009) [1]. Moreover, more explicit representations of such polynomials are given and several classes of explicit linearized polynomials with kernel of any given dimension are presented.