Costas array generator polynomials in finite fields

Permutations of order N are generated using polynomials in a Galois field GF(q) where q ges N+1, which can be written as a linear transformation on a vector of polynomial coefficients. The Lempel and Golomb methods for generating Costas arrays of order q-2 are shown to be very simple examples. The generating polynomial for Costas arrays is examined to form an existence theorem for Costas arrays and a search of polynomial complexity for any given order. Related work is a database on Costas arrays to order 400 and status of an exhaustive search for Costas arrays of order 27.