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 equivalence classes under the dihedral
group that contain a G-symmetric Costas array but no array from 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.; 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.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.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.Rubio,
I. and Torres, J. "Multidimensional Costas Arrays and Their Periodicity."
IEEE Trans. Inform. Th.69, 5032-5040, 2023. https://doi.org/10.1109/TIT.2023.3264951.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).
dihedral
group that contain a G-symmetric Costas array but no array from 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).
