On the degrees of freedom of Costas permutations and other constraints

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August 2011, 5(3): 435-448. doi: 10.3934/amc.2011.5.435

On the degrees of freedom of Costas permutations and other constraints

Konstantinos Drakakis ^1,

1.	School of Electrical, Electronic & Mechanical Engineering, University College Dublin, Belﬁeld, Dublin 4

Received May 2010 Revised January 2011 Published August 2011

Abstract
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The number of degrees of freedom of Costas permutations is considered, and found to be surprisingly small, while partial results about the degrees of freedom of Golomb and Welch Costas permutations are proved. For Golomb Costas permutations, in particular, the curious observation is made that arbitrarily long sequences of distinct positive integers seem to exist, with the property that two or more Golomb Costas permutations, constructed in a suitably large field, start with such a sequence; other types of constraints, related to their cycle structure, are studied; and finally it is shown that, in any extension field containing non-quadratic subfields, Lempel Costas permutations are obtainable through the iterated composition of other Golomb Costas permutations.

Keywords: finite fields, Golomb method, Welch method, Costas arrays/permutations, degrees of freedom, primitive roots..

Mathematics Subject Classification: Primary: 05A05, 11B50; Secondary: 05B1.

Citation: Konstantinos Drakakis. On the degrees of freedom of Costas permutations and other constraints. Advances in Mathematics of Communications, 2011, 5 (3) : 435-448. doi: 10.3934/amc.2011.5.435

References:

[1]	J. K. Beard, Generating Costas Arrays to Order 200, in "Conference on Information Sciences and Systems (CISS) 2006''; availabl online at http://jameskbeard.com/jameskbeard/
[2]	J. K. Beard, Announcement in Conference on Information Sciences and Systems (CISS) 2010.
[3]	C. A. Charalambides, "Enumerative Combinatorics,'' Chapman & Hall/CRC, 2002.
[4]	J. P. Costas, Medium constraints on sonar design and performance, Technical Report Class 1 Rep. R65EMH33, GE Co., 1965.
[5]	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.
[6]	K. Drakakis, A review of Costas arrays, J. Appl. Math., 2006, 32 pp.
[7]	K. Drakakis, A review of the available construction methods for Golomb rulers, Adv. Math. Commun., 3 (2009), 235-250. doi: 10.3934/amc.2009.3.235.
[8]	K. Drakakis, R. Gow and L. O'Carroll, On the symmetry of Welch- and Golomb-constructed Costas arrays, Disc. Math., 309 (2009), 2559-2563. doi: 10.1016/j.disc.2008.04.058.
[9]	K. Drakakis, R. Gow and S. Rickard, On the disjointness of algebraically constructed Costas arrays, J. Algebra Appl., 10 (2011), 219-240. doi: 10.1142/S0219498811004537.
[10]	K. Drakakis, F. Iorio and S. Rickard, The enumeration of Costas arrays of order 28, Adv. Math. Commun., 5 (2011), 69-86. doi: 10.3934/amc.2011.5.69.
[11]	K. Drakakis, F. Iorio, S. Rickard and J. Walsh, Results of the enumeration of Costas arrays of order 29, Adv. Math. Commun., 5 (2011), 547-553.
[12]	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.
[13]	S. Golomb, Algebraic constructions for Costas arrays, J. Combin. Theory Ser. A, 37 (1984), 13-21. doi: 10.1016/0097-3165(84)90015-3.
[14]	S. Golomb and G. Gong, The status of Costas arrays, IEEE Trans. Inform. Theory, 53 (2007), 4260-4265. doi: 10.1109/TIT.2007.907524.
[15]	S. Golomb and H. Taylor, Constructions and properties of Costas arrays, Proc. IEEE, 72 (1984), 1143-1163. doi: 10.1109/PROC.1984.12994.
[16]	G. H. Hardy and E. M. Wright, "An Introduction to the Theory of Numbers,'' 6^th edition, Clarendon Press, Oxford, 2008.
[17]	P. Ribenboim, "The New Book of Prime Number Records,'' 3^rd edition, Springer-Verlag, New York, 1995.
[18]	J. Silverman, V. Vickers and J. Mooney, On the number of Costas arrays as a function of array size, Proc. IEEE, 76 (1988), 851-853. doi: 10.1109/5.7156.

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References:

[1]	J. K. Beard, Generating Costas Arrays to Order 200, in "Conference on Information Sciences and Systems (CISS) 2006''; availabl online at http://jameskbeard.com/jameskbeard/
[2]	J. K. Beard, Announcement in Conference on Information Sciences and Systems (CISS) 2010.
[3]	C. A. Charalambides, "Enumerative Combinatorics,'' Chapman & Hall/CRC, 2002.
[4]	J. P. Costas, Medium constraints on sonar design and performance, Technical Report Class 1 Rep. R65EMH33, GE Co., 1965.
[5]	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.
[6]	K. Drakakis, A review of Costas arrays, J. Appl. Math., 2006, 32 pp.
[7]	K. Drakakis, A review of the available construction methods for Golomb rulers, Adv. Math. Commun., 3 (2009), 235-250. doi: 10.3934/amc.2009.3.235.
[8]	K. Drakakis, R. Gow and L. O'Carroll, On the symmetry of Welch- and Golomb-constructed Costas arrays, Disc. Math., 309 (2009), 2559-2563. doi: 10.1016/j.disc.2008.04.058.
[9]	K. Drakakis, R. Gow and S. Rickard, On the disjointness of algebraically constructed Costas arrays, J. Algebra Appl., 10 (2011), 219-240. doi: 10.1142/S0219498811004537.
[10]	K. Drakakis, F. Iorio and S. Rickard, The enumeration of Costas arrays of order 28, Adv. Math. Commun., 5 (2011), 69-86. doi: 10.3934/amc.2011.5.69.
[11]	K. Drakakis, F. Iorio, S. Rickard and J. Walsh, Results of the enumeration of Costas arrays of order 29, Adv. Math. Commun., 5 (2011), 547-553.
[12]	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.
[13]	S. Golomb, Algebraic constructions for Costas arrays, J. Combin. Theory Ser. A, 37 (1984), 13-21. doi: 10.1016/0097-3165(84)90015-3.
[14]	S. Golomb and G. Gong, The status of Costas arrays, IEEE Trans. Inform. Theory, 53 (2007), 4260-4265. doi: 10.1109/TIT.2007.907524.
[15]	S. Golomb and H. Taylor, Constructions and properties of Costas arrays, Proc. IEEE, 72 (1984), 1143-1163. doi: 10.1109/PROC.1984.12994.
[16]	G. H. Hardy and E. M. Wright, "An Introduction to the Theory of Numbers,'' 6^th edition, Clarendon Press, Oxford, 2008.
[17]	P. Ribenboim, "The New Book of Prime Number Records,'' 3^rd edition, Springer-Verlag, New York, 1995.
[18]	J. Silverman, V. Vickers and J. Mooney, On the number of Costas arrays as a function of array size, Proc. IEEE, 76 (1988), 851-853. doi: 10.1109/5.7156.

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