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To date, two approaches have been followed to identify Costas arrays. One has been exhaustive search, by computer, to find all ntimesn Costas arrays, which has currently been completed for nles26. The other has been the discovery of specific constructions which provide examples of Costas arrays for many different values of n. All the specific construction methods which have been found are related to primitive roots in finite fields, or involve the opportunistic adjoining of an extra "dot", usually at a corner, to go from an example of size n to one of size n+1. For many years, n=32 and n=33 have been the smallest values for which no Costas arrays are known. The last publication that described new specific constructions appeared in 1992. If there are as yet undiscovered specific constructions, one possible way to identify them may be to look at examples found by exhaustive computer search that do not arise from any of the known constructions, and attempt to identify patterns in their formation.