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Costas Array


An order-n Costas array is a permutation on {1,...,n} such that the distances in each row of the triangular difference table are distinct. For example, the permutation {1,3,4,2,5} has triangular difference table {2,1,-2,3}, {3,-1,1}, {1,2}, and {4}. Since each row contains no duplications, the permutation is therefore a Costas array.

There is no known formula, recursion, or generating function for giving the number of Costas arrays of order n. Several number-theoretic generators are known (Golomb 1984, Beard et al. 2004), but these do not generate all known Costas arrays of orders greater than ∼12.

The numbers of n×n Costas arrays for n=1, 2, ... counting flipped and rotated matrices distinctly are 1, 2, 4, 12, 40,116, 200, 444, 760, 2160, 4368, 7852, 12828, 17252, 19612, 21104, 18276, 15096, 10240, 6464, 3536, 2052, 872, 200, 88, 56, 204, ... (OEIS A008404). Here, counts for n=24, 25, and 26 were found by Beard et al. (2004, 2007). n=26 was verified by Rickard et al. (2006) and the case n=27 was solved by Drakakis et al. (2008).

The following table summarizes the order-n Costas arrays for small n.

n#Costas arrays
11(1)
22(1, 2), (2,1)
34(1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2)
412(1, 2, 4, 3), (1, 3, 4, 2), (1, 4, 2, 3), (2, 1, 3, 4), (2, 3, 1, 4), (2, 4, 3, 1),
(3, 1, 2, 4), (3, 2, 4, 1), (3, 4, 2, 1), (4, 1, 3, 2), (4, 2, 1, 3), (4, 3, 1, 2)

The numbers of G-symmetric Costas arrays of order n that are inequivalent under dihedral group but are not given by the Welch construction for n=1, 2, ... are 0, 0, 0, 0, 1, 1, 0, 3, 0, ... (OEIS A008403; apparently given erroneously in Zwillinger 1995, p. 227).


See also

Golomb Ruler

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References

Beard, J. K.; Russo, J. C.; Erickson, K. G.; Monteleone, M. C.; and Wright, M. T. "Combinatoric Collaboration on Costas Arrays and Radar Applications." In Proceedings of the IEEE 2004 Radar Conference, April 26-29 2004.078038234X pp. 260-265, 2004.Beard, J. K.; Russo, J. C.; Erickson, K. G.; Monteleone, M. C.; and Wright, M. T. "Costas Array Generation and Search Methodology." IEEE Trans. Aerospace and Electronic Engineering 43, 522-538, 2007.Costas, J. P. "Medium Constraints on Sonar Design and Performance." General Electric Company Tech. Rep. Class 1 Rep. R65EMH33, Nov. 1965.Costas, J. P. "A Study of Detection Waveforms Having Nearly Ideal Range-Doppler Ambiguity Properties." Proc. IEEE 72, 996-1009, 1984.Drakakis, K.; Rickard, S; Caballero, R.; Iorio, F; O'Brien, G.; and Walsh J. "Results of the Enumeration of Costas Arrays of Order 27." May 23, 2008. http://www.costasarrays.org/Enumeration27TalkWeb.pdf.Golomb, S. W. and Taylor, H. "Construction and Properties of Costas Arrays." Proc. IEEE 72, 1143-1163, 1984.Rickard, S.; Connell, E.; Duignan, F.; Ladendorf, B.; Wade, A. "The Enumeration of Costas Arrays of Size 26." In 2006 40th Annual Conference on Information Sciences and Systems. Princeton, NJ: pp. 815-817, 2006.Silverman, J.; Vickers, V. E.; and Mooney, J. M. "On the Number of Costas Arrays as a Function of Array Size." Proc. IEEE 76, 851-853, 1988.Sloane, N. J. A. Sequences A008403 and A008404 in "The On-Line Encyclopedia of Integer Sequences."Song, H. Y. and Golomb, S. W. "Generalized Welch-Costas Sequences and Their Application to Vatican Arrays." Contemp. Math. 168, 341-351, 1994.Taylor, H. "Costas Arrays." §4.7.6 in CRC Handbook of Combinatorial Designs (Ed. C. J. Colbourn and J. H. Dinitz). Boca Raton, FL: CRC Press, p. 259, 1996.Zwillinger, D. (Ed.). "Costas Arrays." §3.8 in CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 227, 1995.

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Costas Array

Cite this as:

Weisstein, Eric W. "Costas Array." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CostasArray.html

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