Costas Arrays

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Costas arrays are special permutation matrices. The special property is that the distance between any pair of ones in the matrix is not repeated for another pair of ones. This property was originally defined to make the permutation matrix an optimal scheme for setting frequencies in a multiple-tone sonar waveform because it means that unless the receiver is locked on the signal in both frequency and time, no more than one tone will be where it is expected. This property also makes Costas arrays ideal for one of the techniques in sophisticated communications and radar waveforms. For more information, download one or more of the papers below and read the introduction.

For files of data from Costas array research, click here.

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Papers

Jon C. Russo, Keith G. Erickson, and James K Beard, Costas array search technique that maximizes backtrack and symmetry exploitation, CISS 2010, March 17, 2010, Princeton University, Session WA-04, Paper 6; links: paper; slides; addendum slides; CISS even years at Princeton, CISS odd years at JHU. Database of Costas arrays on a CD-ROM given out at CISS 2006 was extended to order 500 and data extraction utility extended to provide generator polynomials in Galois fields (updates to come); see Files page.

Konstantinos Drakakis, Scott Rickard, James K Beard, Rodrigo Caballero, Francesco Iorio, Gareth O'Brien and John Walsh, Results of the enumeration of Costas arrays of order 27, IEEE Transactions on Information Theory 54 10 (October 2008) pp 4684-4687. [The Drakakis group at University College, Dublin, Ireland, announces completion of an exhaustive search over order 27 in this paper. They made their announcement to the community by email on May 23, 2008 but did not notice the only new Costas array of that order in a log file dated March 9; the group of Beard, Russo, Erickson, Monteleone and Wright announced this new Costas array a few days after the Konstantinos group made their announcement; their log file indicates that this array appeared in their logs on April 7 in a processor node provided by Keith Erickson. The Beard team announced the discovery of this Costas array by email on May 29, 2008. See below for the new Costas array.]

Beard, J. K., Costas array generator polynomials in finite fields, CISS 2008, March 21 2008, Princeton University, Session TP 03, Paper 5; links: paper; slides.

Beard, J. K., Russo, J. C., Erickson, K., Monteleone, M., and Wright, M., Costas Array Generation and Search Methodology, IEEE Transactions on Aerospace and Electronic Systems, 43, 2 (April 2007), 522-538. Click here for PDF of scanned paper. Click here for pre-publication draft from original.

Beard, J. K., Generating Costas Arrays to Order 200, presented at Conference on Information Sciences and Systems (CISS) 2006, March 23, 2006. Click here for CD-ROM image with a database of 663,702 Costas arrays of orders 2 through 200. Database extraction utility update is here. Please email me to request a CD-ROM by mail. See Files page for extension of database to order 400 and updates to the data extraction utility.

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Files for Download

A page with links for files useful in working with Costas arrays is here.

New Costas Array of Order 27

The new Costas array of order 27, and its seven polymorphs (rotations, transpositions) is

11 10 4 24 7 23 3 18 21 9 26 16 5 1 15 27 2 25 17 22 19 6 8 12 20 13 14
12 17 10 24 22 8 19 3 7 20 9 16 13 1 2 4 27 26 18 5 23 6 15 25 21 11 14
14 11 21 25 15 6 23 5 18 26 27 4 2 1 13 16 9 20 7 3 19 8 22 24 10 17 12
14 13 20 12 8 6 19 22 17 25 2 27 15 1 5 16 26 9 21 18 3 23 7 24 4 10 11
14 15 8 16 20 22 9 6 11 3 26 1 13 27 23 12 2 19 7 10 25 5 21 4 24 18 17
14 17 7 3 13 22 5 23 10 2 1 24 26 27 15 12 19 8 21 25 9 20 6 4 18 11 16
16 11 18 4 6 20 9 25 21 8 19 12 15 27 26 24 1 2 10 23 5 22 13 3 7 17 14
17 18 24 4 21 5 25 10 7 19 2 12 23 27 13 1 26 3 11 6 9 22 20 16 8 15 14

This is how it appears as a matrix:

                          1                          
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            1                                        
    1                                                
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  1                                                  
1                                                    
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