The concept of (n,k) permutation polynomial over GF(q) is first introduced.The properties of (n,k) permutation polynomials and the relation to k^th order correlationimmune functions have been studied. Sufficient and necessary conditions are proved forsome special n-ary functions to be m^{th(m＜n) order correlation immune and all functions with degree no greater than 2 to be (n－1)thorder correlation immune. The results show that over GF(q)(q＞4) are there nonlinear functions of highest possible correlation immunity order. An efficient method is put forward to construct functions of high nonlinearityfrom those of lower nonlinearity with the same correlation immunity order.}