Costas array: Difference between revisions

Revision as of 06:55, 23 March 2017

Revision as of 01:30, 24 March 2017

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==Known Arrays==

All Costas array orders are known for orders 1 through 29<ref name="JKB200">James K Beard, ''Generating Costas Arrays to Order 200'', 2006 40th Annual Conference on Information Sciences and Systems, (CISS) 2006, March 23, 2006, [https://doi.org/10.1109/ciss.2006.286635| DOI: 10.1109/CISS.2006.286635]</ref><ref>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, Volume: 54, Issue: 10, Oct. 2008, [https://doi.org/10.1109/TIT.2008.928979| DOI: 10.1109/TIT.2008.928979]</ref><ref>K Drakakis, F Iorio, S Rickard, ''The enumeration of Costas arrays of order 28 and its consequences'', Adv. in Math. of Comm., 2011</ref><ref>K Drakakis, F Iorio, S Rickard, J Walsh, ''Results of the Enumeration Of Costas Arrays Of Order 29'', - Adv. in Math. of Comm., Volume 5, No. 3, 2011, 547–553, [https://dx.doi.org/10.3934/amc.2011.5.547| DOI: 10.3934/amc.2011.5.547]</ref> Enumeration is as in the following table.

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Enumeration of known Costas arrays to order 200,<ref name="JKB200"/>, order 500<ref>James K Beard, ''Costas array generator polynomials in finite fields'', 42nd Annual Conference on Information Sciences and Systems (CISS 2008), April 20, 2008, [https://doi.org/10.1109/CISS.2008.4558709| DOI: 10.1109/CISS.2008.455870]</ref> and to order 1030 <ref>http://jameskbeard.com/jameskbeard/Files.html#CostasArrays</ref> are available. Although these lists and databases of these Costas arrays are likely near complete, other Costas arrays with orders above 29 that are not in these lists may exist.

==Constructions==

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Generation of new Costas arrays by adding or subtracting a row/column or two with a 1 or a pair of 1's in a corner were published in a paper focused on generation methods<ref>Solomon Golomb, ''Algebraic constructions for Costas arrays'', J. Comb. Theory Series A, volume 7 (1984), pp 1143-1163</ref> and in Golomb and Taylor's landmark 1984 paper{{sfnp|Golomb|Taylor|1984}}

More sophisticated methods of generating new Costas arrays by deleting rows and columns of existing Costas arrays that were generated by the Welch, Lempel or Golomb generators were published in 1992.<ref>Solomon W. Golomb, ''The T_4and G_4 Constructions for Costas Arrays'', IEEE Transactions on Information Theory, volume 38 (1992), pp 1404-1406.</ref>. There is no upper limit on the order for which these generators will produce Costas arrays.

===Other Methods===

Two methods that found Costas arrays up to order 52 using more complicated methods of adding or deleting rows and columns were published in 2004<ref>Scott Rickard, ''Searching for Costas Arrays using Periodicity Properties'', IMA International Conference on Mathematics in Signal Processing (2004})</ref> and 2007<ref>James K. Beard, Jon C. Russo and Keith G. Erickson and Michael C. Monteleone and Michael T. Wright, ''Costas array generation and search methodology'', IEEE Transactions on Aerospace and Electronic Systems, volume 43 number 2, April 2007, pp 522-538, [https://doi.org/10.1109/TAES.2007.4285351| DOI: 10.1109/TAES.2007.4285351]</ref>

== See also ==