A Costas array of order ,
where
is a positive integer, is a permutation of such that, for each with , the signed differences for are pairwise distinct. For example, the permutation has triangular difference
table ,
, , and . Since each row contains no duplications, the permutation
is therefore a Costas array.
Several algebraic construction methods are known (Golomb and Taylor 1984, Beard et al. 2004), but they do not generate every Costas array in general. Order 28 is
an exception for which every Costas array is accounted for by the known algebraic
constructions (Drakakis et al. 2011).
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 .
A Costas array corresponding to a permutation is called G-symmetric if, for even , for . For odd , the conditions are and for (Jedwab and Wodlinger 2014, Definition 4.8).
The Welch construction gives a Costas array of order by taking to be the least positive residue of modulo for , where is a prime number, is a primitive root modulo
, and (Jedwab and Wodlinger 2014, Theorem 4.14).
The numbers of G-symmetric Costas arrays of order that are inequivalent under the 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. "Costas Array Generation and
Search Methodology." IEEE Trans. Aerospace Electr. Systems43,
522-538, 2007. https://doi.org/10.1109/TAES.2007.4285351.Beard,
J. K.; Russo, J. C.; Erickson, K.; Monteleone, M.; and Wright, M. "Combinatoric
Collaboration on Costas Arrays and Radar Applications." In Proceedings of
the IEEE 2004 Radar Conference. Piscataway, NJ: IEEE, pp. 260-265, 2004.
https://doi.org/10.1109/NRC.2004.1316432.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.; Iorio, F.; and Rickard, S. "The Enumeration of Costas Arrays of Order 28
and Its Consequences." Adv. Math. Commun.5, 69-86, 2011. https://doi.org/10.3934/amc.2011.5.69.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.Jedwab, J. and Wodlinger,
J. "Structural Properties of Costas Arrays." Adv. Math. Commun.8,
241-256, 2014. https://doi.org/10.3934/amc.2014.8.241.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. https://doi.org/10.1109/CISS.2006.286579.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.
,
where 
is a positive integer, is a permutation 
of 
such that, for each 
with 
, the signed differences 
for 
are pairwise distinct. For example, the permutation 
has triangular difference
table 
,
, 
, and 
. Since each row contains no duplications, the permutation
is therefore a Costas array.
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 
.
is called G-symmetric if, for even 
, 
for 
. For odd 
, the conditions are 
and 
for 
(Jedwab and Wodlinger 2014, Definition 4.8).
The Welch construction gives a Costas array of order 
by taking 
to be the least positive residue of 
modulo 
for 
, where 
is a prime number, 
is a primitive root modulo
, and 
(Jedwab and Wodlinger 2014, Theorem 4.14).
that are inequivalent under the 
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).
