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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

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 fi elds, 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/ Google Scholar

[2]

J. K. Beard, Announcement in Conference on Information Sciences and Systems (CISS) 2010. Google Scholar

[3]

C. A. Charalambides, "Enumerative Combinatorics,'' Chapman & Hall/CRC, 2002.  Google Scholar

[4]

J. P. Costas, Medium constraints on sonar design and performance, Technical Report Class 1 Rep. R65EMH33, GE Co., 1965. Google Scholar

[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.  Google Scholar

[6]

K. Drakakis, A review of Costas arrays,, J. Appl. Math., 2006 ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

S. Golomb and H. Taylor, Constructions and properties of Costas arrays, Proc. IEEE, 72 (1984), 1143-1163. doi: 10.1109/PROC.1984.12994.  Google Scholar

[16]

G. H. Hardy and E. M. Wright, "An Introduction to the Theory of Numbers,'' 6th edition, Clarendon Press, Oxford, 2008.  Google Scholar

[17]

P. Ribenboim, "The New Book of Prime Number Records,'' 3rd edition, Springer-Verlag, New York, 1995.  Google Scholar

[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.  Google Scholar

show all references

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/ Google Scholar

[2]

J. K. Beard, Announcement in Conference on Information Sciences and Systems (CISS) 2010. Google Scholar

[3]

C. A. Charalambides, "Enumerative Combinatorics,'' Chapman & Hall/CRC, 2002.  Google Scholar

[4]

J. P. Costas, Medium constraints on sonar design and performance, Technical Report Class 1 Rep. R65EMH33, GE Co., 1965. Google Scholar

[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.  Google Scholar

[6]

K. Drakakis, A review of Costas arrays,, J. Appl. Math., 2006 ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

S. Golomb and H. Taylor, Constructions and properties of Costas arrays, Proc. IEEE, 72 (1984), 1143-1163. doi: 10.1109/PROC.1984.12994.  Google Scholar

[16]

G. H. Hardy and E. M. Wright, "An Introduction to the Theory of Numbers,'' 6th edition, Clarendon Press, Oxford, 2008.  Google Scholar

[17]

P. Ribenboim, "The New Book of Prime Number Records,'' 3rd edition, Springer-Verlag, New York, 1995.  Google Scholar

[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.  Google Scholar

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