1,2

A Costas array is a permutation matrix that meets the Costas condition. The Costas condition has several equivalent definitions. One of them is that two square matrices defined from a Costas array, when overlaid with one of them offset by an integral number of rows and columns, will have no more than one 1 overlaid on another except when the number of shifts in both rows and columns is zero. - James K Beard (jkbeard(AT)ieee.org), Nov 07 2005

CRC Handbook of Combinatorial Designs, C. Colbourn and J. Dinitz Eds., 1996, IV.7: Costas Arrays by Herbert Taylor (IV.7.6, page 259, Table 2.29).

CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 227.

J. Silverman, V. E. Vickers and J. M. Mooney, On the number of Costas arrays as a function of array size, Proc. IEEE, 76 (1988), 851-853.

James K Beard, Jon C Russo, Keith Erickson, Michael Moneleone and Mike Wright, Combinatoric collaboration on Costas arrays and radar applications, Proceedings of the IEEE 2004 Radar Conference, Apr 26, 2004, ISBN 0-7803-8234-X, pp. 260-265 (entries for orders 24 and 25).

James K Beard, Jon C Russo, Keith Erickson, Michael Moneleone and Mike Wright,"Costas Array Generation and Search Methodology," to appear in IEEE Transactions on Aerospace and Electronic Engineering. (Order 26)

Ed Pegg, Jr., Golomb Rulers

Eric Weisstein's World of Mathematics, Costas Array

K. Drakakis, Results of the enumeration of Costas arrays oforder 27.

There is no formula, recursion, or generating function for Costas arrays. A number of number-theoretic generators are known (see Golomb 1984, Beard 2004, etc.) but these do not generate all known Costas arrays of orders greater than twelve or so. - James K Beard (jkbeard(AT)ieee.org), Nov 07 2005

A permutation matrix can be represented by a sequence of column indices, one for each row. A previously unknown Costas array of order 26 given this way is

(5, 8, 20, 16, 18, 15, 4, 25, 13, 19, 6, 10, 2, 0, 9, 24, 14, 21, 3, 23, 22, 7, 1, 11, 12, 17)

We use backtrack programming for exhaustive search and number-theoretic generators for the Costas arrays that can be found that way. See Beard et al., 2004 and IEEE Transacations AES, to appear.

Cf. A001441.

Sequence in context: A149846 A108532 A000940 this_sequence A099214 A126946 A113179

Adjacent sequences: A008401 A008402 A008403 this_sequence A008405 A008406 A008407

nonn

N. J. A. Sloane (njas(AT)research.att.com).

More terms from James K Beard (jkbeard(AT)ieee.org), Nov 07 2005

a(27) (from the Drakakis link) sent by John Healy (johnjhealy(AT)gmail.com), Jul 17 2008