The authors investigate a certain class of permutation polynomials of the following form. Let \(F= \mathrm{GF}(q)\) and \(L(x)\in F[x]\) be a \(\mathrm{GF}(q)\)-linear polynomial. Then they consider polynomials of the form \(L(x)x^k\) such that these are permutation polynomials. For example \(q=2\), \(n\) odd, \(1< m< n\) relatively prime to \(n,k= 2A^{n-1}- 2^{m-1}- 1\) and \[ L(x)= \sum^{m-1}_{i=0} x^{2^i}. \] Then \(L(x)x^k\) is such a permutation polynomial. Using these polynomials one can define a quasifield multiplication on \(F\). This is used by the authors to construct translation planes of order \(|F|\). These planes are new, admit a cyclic collineation group of order \(|F|- 1\) fixing two points at the line of infinity and permuting the remaining points transitively.