Journal of Algebra and Its Applications Vol. 10, No. 2 (2011) 219–240 c  World Scientific Publishing Company DOI: 10.1142/S0219498811004537 ON THE DISJOINTNESS OF ALGEBRAICALLY CONSTRUCTED COSTAS ARRAYS KONSTANTINOS DRAKAKIS ∗,†,‡,¶ , ROD GOW ∗,‡,§,‖ and SCOTT RICKARD ∗,†,‡,∗∗ ∗ UCD CASL, University College Dublin Belfield, Dublin 4, Ireland † School of Electrical, Electronic, and Mechanical Engineering University College Dublin, Belfield, Dublin 4, Ireland ‡ Claude Shannon Institue, Ireland § School of Mathematics, University College Dublin Belfield, Dublin 4, Ireland ¶ konstantinos.drakakis@ucd.ie ‖ rod.gow@ucd.ie ∗∗ scott.rickard@ucd.ie Received 7 March 2008 Accepted 29 April 2010 Communicated by J. Rosenthal Is it possible for a particular Costas array to be generated by two different constructions of the Golomb and Welch families? Experimental data suggests that this does not happen (except for trivially small orders), and a (partial) proof of this fact is offered herein through a case-by-case study of all possible pairs of constructions that can potentially produce Costas arrays of equal order. Keywords : Costas arrays; construction techniques; Golomb construction; Lempel construction; Welch construction; optimal autocorrelation sequences. 1. Introduction Construction of new Costas arrays [5, 6] today is based almost exclusively on the two algebraic methods based on finite fields and published in 1984, known as the Golomb and the Welch technique [8, 18, 22]. In the absence of a complete mathe- matical characterization of the Costas property itself, exhaustive search is still the only method guaranteed to yield all Costas arrays for a given order n, but, as all currently known algorithms implementing this search have exponential complexity in n, application to large orders is prohibitive; as of this date exhaustive search has enumerated all Costas arrays up to and including order 28 (for a history of Costas arrays enumerations see, for example, [1, 2, 14, 15, 25], and references therein). Analysis of these (rather limited) enumeration results available today shows that 219