An order-
Costas array is a permutation on such that the distances in each row of the triangular
difference table are distinct. For example, the permutation has triangular difference table , , , and . 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 .
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 .
The numbers of
Costas arrays for ,
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 ,
25, and 26 were found by Beard et al. (2004, 2007). was verified by Rickard et al. (2006) and the case
was solved by Drakakis et al.
(2008).
The following table summarizes the order- Costas arrays for small .
The numbers of G-symmetric Costas arrays of order that are inequivalent under dihedral group but are not given
by the Welch construction for , 2, ... are 0, 0, 0, 0, 1, 1, 0, 3, 0, ... (OEIS A008403;
apparently given erroneously in Zwillinger 1995, p. 227).
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 Engineering43, 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. IEEE72, 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. IEEE72, 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. IEEE76, 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.
Costas array is a permutation on 
such that the distances in each row of the triangular
difference table are distinct. For example, the permutation 
has triangular difference table 
, 
, 
, and 
. Since each row contains no duplications, the permutation
is therefore a Costas array.
.
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 
.
Costas arrays for 
,
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 
,
25, and 26 were found by Beard et al. (2004, 2007). 
was verified by Rickard et al. (2006) and the case
was solved by Drakakis et al.
(2008).
Costas arrays for small 
.
that are inequivalent under dihedral group but are not given
by the Welch construction for 
, 2, ... are 0, 0, 0, 0, 1, 1, 0, 3, 0, ... (OEIS A008403;
apparently given erroneously in Zwillinger 1995, p. 227).
