This page is a clearinghouse for information concerning Costas Arrays.

In the 1960’s, Dr. John P. Costas, motivated by a novel SONAR application, began searching for permutation matrices with ideal auto-ambiguity properties. By hand, he found examples of such matrices of size up to N = 12. Unable to find one of size 13, he contacted Professor Solomon Golomb who then provided generation techniques based on the theory of finite fields for creating these matrices, dubbed Costas arrays. The generation methods produce Costas arrays for infinitely many N, but not all N. For example, the techniques can be used to generate arrays for all N ≤ 31, but no Costas array of size N = 32 or N = 33 has been found. Computer search has enumerated all Costas arrays of size N ≤ 29, but the exponential growth of the search space prohibits extending these results much further with current computational capabilities. After nearly 40 years of research, the first question concerning Costas arrays remains open:

Do Costas arrays exist for all N?

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- C(N) = number of Costas Arrays of size N.
- c(N) = number of unique (up to rotation and reflection) Costas Arrays of size N.

N | C(N) | c(N) |
---|---|---|

1 | 1 | 1 |

2 | 2 | 1 |

3 | 4 | 1 |

4 | 12 | 2 |

5 | 40 | 6 |

6 | 116 | 17 |

7 | 200 | 30 |

8 | 444 | 60 |

9 | 760 | 100 |

10 | 2160 | 277 |

11 | 4368 | 555 |

12 | 7852 | 990 |

13 | 12828 | 1616 |

14 | 17252 | 2168 |

15 | 19612 | 2467 |

16 | 21104 | 2648 |

17 | 18276 | 2294 |

18 | 15096 | 1892 |

19 | 10240 | 1283 |

20 | 6464 | 810 |

21 | 3536 | 446 |

22 | 2052 | 259 |

23 | 872 | 114 |

24 | 200 | 25 |

25 | 88 | 12 |

26 | 56 | 8 |

27 | 204 | 29 |

28 | 712 | 89 |

29 | 164 | 23 |

- A file which contains all Costas Arrays of size 1 to 27 is available here: http://osl-vps-4.ucd.ie/downloader
- The format is, for example,
- c10all.out contains all 10-by-10 Costas Arrays.
- c10uni.out contains one member (the first lexicographically) from each dihedral equivalence class for all the 10-by-10 Costas Arrays.

- Costas database of paper is here:
- The key papers would include:
- Original Costas paper:

http://www.costasarrays.org/costasrefs/costas84study.pdf

- Good place to start is Golomb&Taylor 1984:

http://www.costasarrays.org/costasrefs/golomb84constructions.pdf - Key paper about cross-correlation:

http://www.costasarrays.org/costasrefs/freedman85any.pdf - Nice observation about definition of Costas array:

http://www.costasarrays.org/costasrefs/chang87remark.pdf - Silverman et. al. paper on enumeration of Costas arrays:

http://www.costasarrays.org/costasrefs/silverman88number.pdf - Rickard's MS Thesis

http://www.costasarrays.org/costasrefs/rickard93large.pdf - Rickard's discovery of 'new' arrays:

http://www.costasarrays.org/costasrefs/rickard04searching.pdf - Golomb's view on the current status:

http://www.costasarrays.org/costasrefs/golomb06status.pdf - Drakakis' review of the current status:

http://www.costasarrays.org/costasrefs/drakakis06review.pdf - Herbert Taylor and Jeffrey H. Dinitz' excellent book chapter on Costas arrays:

Herbert Taylor, Jeffrey H. Dinitz, Part IV, Chapter 9 - Costas Arrays (pages 357-361) in*Handbook of Combinatorial Designs, Second Edition,*Edited by Charles J. Colbourn, Jeffrey H. Dinitz, Chapman & Hall/CRC, 2006.

- Original Costas paper:
- Talks:
- Rickard's IMA 2006 slides with some open problems:

http://www.costasarrays.org/costasrefs/rickard06open.pdf - The results of the enumeration of C(27):

http://www.costasarrays.org/Enumeration27TalkWeb.pdf

- Rickard's IMA 2006 slides with some open problems:

- Available now. Please see the Costas Array MATLAB Toolbox page: http://osl-vps-4.ucd.ie/

- Mathworld: http://mathworld.wolfram.com/CostasArray.html
- Wikipedia: http://en.wikipedia.org/wiki/Costas_array
- Mathematical Association of America: http://www.maa.org/editorial/mathgames/mathgames_11_15_04.html
- Claude Shannon Institute: http://www.shannoninstitute.ie/html/costas.html
- The On-Line Encyclopedia of Integer Sequences: http://www.research.att.com/~njas/sequences/A008404

- Please send comments/suggestions/corrections to:
- Scott Rickard (http://sparse.ucd.ie/)
- scott dot rickard - at - ucd dot ie
- Last Update: 04-Oct-2011 11:45 PM