Costas array: Difference between revisions

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Enumeration of known Costas arrays to order 200,{{sfnp|Beard|2006}} order 500<ref>James K {{sfnp|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>{{harvtxt|Beard|2017}}; {{citation|url=http://jameskbeard.com/jameskbeard/Files.html#CostasArrays</ref><ref>James|title=Files Kfor Beard,Download: "Costas arraysArrays|first=James and enumeration to order 1030", IEEE Dataport, 2017K.|last=Beard|accessdate=2020-04-20}} [Online]. Available: https://dx.doi.org/10.21227/H21P42. Accessed: Sept. 17, 2017.</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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===Extensions by Taylor, Lempel, and Golomb===

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 {{sfnp|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. {{sfnp|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 {{sfnp|Rickard, ''Searching for Costas Arrays using Periodicity Properties'', IMA International Conference on Mathematics in Signal Processing (|2004)</ref>}} and 2007.<ref>James K. {{sfnp|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 ==

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| publisher = IEEE

| title = 2006 40th Annual Conference on Information Sciences and Systems}}.

*{{citation

| last = Beard | first = James K.

| contribution = Costas array generator polynomials in finite fields

| date = March 2008

| doi = 10.1109/ciss.2008.4558709

| publisher = IEEE

| title = 2008 42nd Annual Conference on Information Sciences and Systems}}.

*{{citation

| last = Beard | first = James K.

| doi = 10.21227/H21P42

| publisher = IEEE Dataport

| title = Costas arrays and enumeration to order 1030

| url = https://ieee-dataport.org/open-access/costas-arrays-and-enumeration-order-1030

| year = 2017}}.

*{{citation

| last1 = Beard

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| url-status = dead

}}.

*{{citation

| last1 = Beard | first1 = James

| last2 = Russo | first2 = Jon

| last3 = Erickson | first3 = Keith

| last4 = Monteleone | first4 = Michael

| last5 = Wright | first5 = Michael

| date = April 2007

| doi = 10.1109/taes.2007.4285351

| issue = 2

| journal = IEEE Transactions on Aerospace and Electronic Systems

| pages = 522–538

| title = Costas array generation and search methodology

| first1volume = S. W43}}.

*{{citation

| last = Costas | first = J. P. | author-link = John P. Costas (engineer)

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| year = 1965}}.

*{{citation

| last = Golomb | first = Solomon W. | authorlink = Solomon W. Golomb

| last1 = Golomb

| doi = 10.1016/0097-3165(84)90015-3

| first1 = S. W.

| issue = 1

| author1-link = Solomon W. Golomb

| journal = Journal of Combinatorial Theory | series = Series A

| last2 = Taylor

| first2mr = H.749508

| pages = 13–21

| title = Algebraic constructions for Costas arrays

| volume = 37

| year = 1984}}.

*{{citation

| last = Golomb | first = Solomon W. | authorlink = Solomon W. Golomb

| doi = 10.1109/18.144726

| issue = 4

| journal = IEEE Transactions on Information Theory

| mr = 1168761

| pages = 1404–1406

| title = The <math>T_4</math> and <math>G_4</math> constructions for Costas arrays

| volume = 38

| year = 1992}}

*{{citation

| last1 = Golomb | first1 = S. W. | author1-link = Solomon W. Golomb

| last2 = Taylor | first2 = H.

| doi = 10.1109/PROC.1984.12994

| issue = 9

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| volume = 542

| year = 1999}}.

*{{citation

| last = Rickard | first = Scott

| contribution = Searching for Costas Arrays using Periodicity Properties

| title = IMA International Conference on Mathematics in Signal Processing

| year = 2004}}.

== External links ==