Results of the enumeration of Costas arrays of order 29

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On linear equivalence and Phelps codes. Addendum

August 2011, 5(3): 547-553. doi: 10.3934/amc.2011.5.547

Results of the enumeration of Costas arrays of order 29

Konstantinos Drakakis ^1, , Francesco Iorio ^2, , Scott Rickard ^1, and John Walsh ^3,

1.	School of Electrical, Electronic & Mechanical Engineering, University College Dublin, Belﬁeld, Dublin 4
2.	Autodesk Research, 210 King Street East, Toronto, Ontario M5A 1J7
3.	Department of Computer Science, Trinity College Dublin, College Green, Dublin 2, Ireland

Received January 2011 Published August 2011

Abstract
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The results of the enumeration of Costas arrays of order 29 are presented: except for 16 arrays out of a total of 164, all other arrays found are accounted for by the Golomb and Welch construction methods. These 16 arrays, however, cannot be considered to be new, as they were discovered in the past through a semi-empirical technique. The enumeration was performed on several computer clusters and required the equivalent of 366.55 years of single CPU time.

Keywords: Welch method., order 29, enumeration, Costas arrays, Golomb method.

Mathematics Subject Classification: Primary: 05A15, 05B10; Secondary: 05A05, 11B5.

Citation: Konstantinos Drakakis, Francesco Iorio, Scott Rickard, John Walsh. Results of the enumeration of Costas arrays of order 29. Advances in Mathematics of Communications, 2011, 5 (3) : 547-553. doi: 10.3934/amc.2011.5.547

References:

[1]	J. P. Costas, Medium constraints on SONAR design and performance, in "Technical Report Class 1 Rep. R65EMH33,'' GE Co., 1965; A synopsis of this report appeared in the EASCON Convention Record, (1975), 68A-68L.
[2]	J. P. Costas, A study of detection waveforms having nearly ideal range-doppler ambiguity properties, Proc. IEEE, 72 (1984), 996-1009. doi: 10.1109/PROC.1984.12967.
[3]	K. Drakakis, A review of Costas arrays,, J. Appl. Math., 2006 ().
[4]	K. Drakakis, F. Iorio and S. Rickard, The enumeration of Costas arrays of order 28 and its consequences, Adv. Math. Commun., 5 (2011), 69-86. doi: 10.3934/amc.2011.5.69.
[5]	K. Drakakis, S. Rickard, J. Beard, R. Caballero, F. Iorio, G. O'Brien and J. Walsh, Results of the enumeration of Costas arrays of order 27, IEEE Trans. Inform. Theory, 54 (2008), 4684-4687. doi: 10.1109/TIT.2008.928979.
[6]	S. W. Golomb, Algebraic constructions for Costas arrays, J. Combin. Theory Ser. A, 37 (1984), 13-21. doi: 10.1016/0097-3165(84)90015-3.
[7]	S. W. Golomb, The $T_4$ and $G_4$ constructions for Costas arrays, IEEE Trans. Inform. Theory, 38 (1992), 1404-1406. doi: 10.1109/18.144726.
[8]	S. W. Golomb and H. Taylor, Constructions and properties of Costas arrays, Proc. IEEE, 72 (1984), 1143-1163. doi: 10.1109/PROC.1984.12994.
[9]	S. Rickard, Searching for Costas arrays using periodicity properties, in "IMA International Conference on Mathematics in Signal Processing at The Royal Agricultural College,'' Cirencester, UK, 2004.

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References:

[1]	J. P. Costas, Medium constraints on SONAR design and performance, in "Technical Report Class 1 Rep. R65EMH33,'' GE Co., 1965; A synopsis of this report appeared in the EASCON Convention Record, (1975), 68A-68L.
[2]	J. P. Costas, A study of detection waveforms having nearly ideal range-doppler ambiguity properties, Proc. IEEE, 72 (1984), 996-1009. doi: 10.1109/PROC.1984.12967.
[3]	K. Drakakis, A review of Costas arrays,, J. Appl. Math., 2006 ().
[4]	K. Drakakis, F. Iorio and S. Rickard, The enumeration of Costas arrays of order 28 and its consequences, Adv. Math. Commun., 5 (2011), 69-86. doi: 10.3934/amc.2011.5.69.
[5]	K. Drakakis, S. Rickard, J. Beard, R. Caballero, F. Iorio, G. O'Brien and J. Walsh, Results of the enumeration of Costas arrays of order 27, IEEE Trans. Inform. Theory, 54 (2008), 4684-4687. doi: 10.1109/TIT.2008.928979.
[6]	S. W. Golomb, Algebraic constructions for Costas arrays, J. Combin. Theory Ser. A, 37 (1984), 13-21. doi: 10.1016/0097-3165(84)90015-3.
[7]	S. W. Golomb, The $T_4$ and $G_4$ constructions for Costas arrays, IEEE Trans. Inform. Theory, 38 (1992), 1404-1406. doi: 10.1109/18.144726.
[8]	S. W. Golomb and H. Taylor, Constructions and properties of Costas arrays, Proc. IEEE, 72 (1984), 1143-1163. doi: 10.1109/PROC.1984.12994.
[9]	S. Rickard, Searching for Costas arrays using periodicity properties, in "IMA International Conference on Mathematics in Signal Processing at The Royal Agricultural College,'' Cirencester, UK, 2004.

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