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The triple autocorrelation of a sequence is shown to be a Lempel Costas array iff the sequence is maximal, assuming correlation is appropriately defined and the alphabet of the sequence is of prime size. A method that allows the construction of thumbtack autocorrelation sequences of arbitrary alphabet size is subsequently proposed. [ABSTRACT FROM AUTHOR]

A Costas array is an arrangement of N dots on an N-by-N grid, one per row, one per column, such that no two dots share the same displacement vector with any other pair. Costas arrays have applications in SONAR/RADAR systems, communication systems, cryptography, and other areas. We present a standardization of notation and language which can be used to discuss Costas array generation techniques and array manipulations. Using this standardization we can concisely and clearly state various theorems about Costas arrays, including several new theorems about the symmetries of Costas arrays. We also define labels for each array (generated, emergent, and sporadic), which describe whether the array is generated using a known technique, generated using a semiempirical variation of a known technique, or of unexplained origin. A new method for obtaining emergent Costas arrays, the DRT expansion, is also given here for the first time. A MATLAB Costas array toolbox has also been developed which implements the proposed standardization. The toolbox contains a comprehensive set of functions covering Costas array generation, manipulation and classification. [ABSTRACT FROM AUTHOR]

In this paper, we show that in order to ascertain whether a permutation has the Costas property, only a restricted subset among the totality of pairs of entries in the same row of the difference triangle needs to be checked, and we explicitly describe such a subset. This represents a further refinement on the definition of a Costas permutation. This observation can be used to speed up algorithms that exhaustively search for Costas permutations. Asymptotically, the savings approaches 43% for large orders when compared with the previous standard efficient method. [ABSTRACT FROM AUTHOR]

Costas arrays are not only useful in radar engineering, but they also present many interesting, and still open, mathematical problems. This work collects in it all important knowledge about them available today: some history of the subjects, density results, construction methods, construction algorithms with full proofs, and open questions. At the same time all the necessary mathematical background is offered in the simplest possible format and terms, so that this work can play the role of a reference for mathematicians and mathematically inclined engineers interested in the field. [ABSTRACT FROM AUTHOR]

Costas arrays are used in RADAR (and SONAR) engineering to represent frequency-hopping patterns that optimize the RADAR's performance. In this work, using all available Costas arrays up to and including order 26, as well as data mining techniques, we investigate how the Costas property and the balance of signs in the difference triangle of a permutation are related. Our conclusion is that there is sufficient evidence to believe that the mechanism responsible for the formation of Costas arrays changes from low to high orders. [ABSTRACT FROM AUTHOR]