Costas arrays and the Lovász Local Lemma

We propose a fundamentally new approach to studying the existence of Costas arrays. We explore the consequences of applying new formulations of the Lovász Local Lemma in an effort to apply the probabilistic method to show that Costas arrays of a given order n must exist. Our applications of the Lovász Local Lemma make use of a characterization of all O(n⁶) configurations of ones within a permutation matrix whose existence violates the definition of a Costas array. We show that this approach can at least be used to establish the existence of arrays of orders 3 - 5 and also can be applied to prove the existence of frequency-hopping waveforms of all orders satisfying many subsets of at least n(n - 1)(n - 2)² - 2 of the violating constraints.