# American Institute of Mathematical Sciences

August  2014, 8(3): 241-256. doi: 10.3934/amc.2014.8.241

## Structural properties of Costas arrays

 1 Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC, Canada V5A 1S6 2 Department of Mathematics and Statistics, University of Victoria, 3800 Finnerty Road, Victoria BC V8P 5C2, Canada

Received  August 2012 Revised  March 2014 Published  August 2014

We apply combinatorial arguments to establish structural constraints on Costas arrays. We prove restrictions on when a Costas array can contain a large corner region whose entries are all 0. In particular, we prove a 2010 conjecture due to Russo, Erickson and Beard. We then constrain the vectors joining pairs of 1s in a Costas array by establishing a series of results on its number of "mirror pairs," namely pairs of these vectors having the same length but opposite slopes.
Citation: Jonathan Jedwab, Jane Wodlinger. Structural properties of Costas arrays. Advances in Mathematics of Communications, 2014, 8 (3) : 241-256. doi: 10.3934/amc.2014.8.241
##### References:
 [1] J. K. Beard, Orders three through 100,, Database of Costas arrays, (). Google Scholar [2] S. R. Blackburn, T. Etzion, K. M. Martin and M. B. Paterson, Two-dimensional patterns with distinct differences - constructions, bounds, and maximal anticodes,, IEEE Trans. Inform. Theory, 56 (2010), 1216. doi: 10.1109/TIT.2009.2039046. Google Scholar [3] C. P. Brown, M. Cenkl, R. A. Games, J. Rushanan, O. Moreno and P. Pei, New enumeration results for Costas arrays,, in Proc. IEEE Int. Symp. Inform. Theory, (1993). doi: 10.1109/ISIT.1993.748721. Google Scholar [4] J. P. Costas, A study of a class of detection waveforms having nearly ideal range-Doppler ambiguity properties,, Proc. IEEE, 72 (1984), 996. Google Scholar [5] K. Drakakis, A review of Costas arrays,, J. Appl. Math., (2006). doi: 10.1155/JAM/2006/26385. Google Scholar [6] K. Drakakis, Open problems in Costas arrays,, preprint, (). Google Scholar [7] K. Drakakis, R. Gow, J. Healy and S. Rickard, Cross-correlation properties of Costas arrays and their images under horizontal and vertical flips,, Math. Problems Engin., (2008). doi: 10.1155/2008/369321. Google Scholar [8] K. Drakakis, R. Gow and S. Rickard, Interlaced Costas arrays do not exist,, Math. Problems Engin., (2008). doi: 10.1155/2008/456034. Google Scholar [9] K. Drakakis, R. Gow and S. Rickard, Common distance vectors between Costas arrays,, Adv. Math. Commun., 3 (2009), 35. doi: 10.3934/amc.2009.3.35. Google Scholar [10] K. Drakakis, F. Iorio and S. Rickard, The enumeration of Costas arrays of order 28 and its consequences,, Adv. Math. Commun., 5 (2011), 69. doi: 10.3934/amc.2011.5.69. Google Scholar [11] P. Erdős, R. Graham, I. Z. Ruzsa and H. Taylor, Bounds for arrays of dots with distinct slopes or lengths,, Combinatorica, 12 (1992), 39. doi: 10.1007/BF01191203. Google Scholar [12] P. Erdős and P. Turán, On a problem of Sidon in additive number theory and on some related problems,, J. London Math. Soc., 16 (1941), 212. Google Scholar [13] T. Etzion, Combinatorial designs derived from Costas arrays,, in Sequences: Combinatorics, (1990), 208. Google Scholar [14] T. Etzion, Sequence folding, lattice tiling, and multidimensional coding,, IEEE Trans. Inform. Theory, 57 (2011), 4383. doi: 10.1109/TIT.2011.2146010. Google Scholar [15] A. Freedman and N. Levanon, Any two $N \times N$ Costas arrays must have at least one common ambiguity sidelobe if $N>3$ - a proof,, Proc. IEEE, 73 (1985), 1530. Google Scholar [16] E. N. Gilbert, Latin squares which contain no repeated digrams,, SIAM Review, 7 (1965), 189. doi: 10.1137/1007035. Google Scholar [17] S. Golomb, Algebraic constructions for Costas arrays,, J. Combin. Theory Ser. A, 37 (1984), 13. doi: 10.1016/0097-3165(84)90015-3. Google Scholar [18] S. W. Golomb and H. Taylor, Constructions and properties of Costas arrays,, Proc. IEEE, 72 (1984), 1143. doi: 10.1109/PROC.1984.12994. Google Scholar [19] S. Rickard, Database of Costas arrays,, accessed 2012., (2012). Google Scholar [20] S. Rickard, Searching for Costas arrays using periodicity properties,, in Proc. 2004 IMA Int. Conf. Math. Signal Proc., (2004). Google Scholar [21] J. C. Russo, K. G. Erickson and J. K. Beard, Costas array search technique that maximizes backtrack and symmetry exploitation,, in 44th Annual Conf. Inform. Sci. Systems, (2010). doi: 10.1109/CISS.2010.5464772. Google Scholar [22] H. Taylor, Non-attacking rooks with distinct differences,, Technical Report CSI-84-03-02, (1984), 84. Google Scholar [23] K. Taylor, S. Rickard and K. Drakakis, On Costas arrays with various types of symmetry,, in 16th Int. Conf. Digital Signal Proc., (2009). doi: 10.1109/ICDSP.2009.5201135. Google Scholar [24] J. L. Wodlinger, Costas Arrays, Golomb Rulers and Wavelength Isolation Sequence Pairs,, Master's thesis, (2012). Google Scholar

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##### References:
 [1] J. K. Beard, Orders three through 100,, Database of Costas arrays, (). Google Scholar [2] S. R. Blackburn, T. Etzion, K. M. Martin and M. B. Paterson, Two-dimensional patterns with distinct differences - constructions, bounds, and maximal anticodes,, IEEE Trans. Inform. Theory, 56 (2010), 1216. doi: 10.1109/TIT.2009.2039046. Google Scholar [3] C. P. Brown, M. Cenkl, R. A. Games, J. Rushanan, O. Moreno and P. Pei, New enumeration results for Costas arrays,, in Proc. IEEE Int. Symp. Inform. Theory, (1993). doi: 10.1109/ISIT.1993.748721. Google Scholar [4] J. P. Costas, A study of a class of detection waveforms having nearly ideal range-Doppler ambiguity properties,, Proc. IEEE, 72 (1984), 996. Google Scholar [5] K. Drakakis, A review of Costas arrays,, J. Appl. Math., (2006). doi: 10.1155/JAM/2006/26385. Google Scholar [6] K. Drakakis, Open problems in Costas arrays,, preprint, (). Google Scholar [7] K. Drakakis, R. Gow, J. Healy and S. Rickard, Cross-correlation properties of Costas arrays and their images under horizontal and vertical flips,, Math. Problems Engin., (2008). doi: 10.1155/2008/369321. Google Scholar [8] K. Drakakis, R. Gow and S. Rickard, Interlaced Costas arrays do not exist,, Math. Problems Engin., (2008). doi: 10.1155/2008/456034. Google Scholar [9] K. Drakakis, R. Gow and S. Rickard, Common distance vectors between Costas arrays,, Adv. Math. Commun., 3 (2009), 35. doi: 10.3934/amc.2009.3.35. Google Scholar [10] K. Drakakis, F. Iorio and S. Rickard, The enumeration of Costas arrays of order 28 and its consequences,, Adv. Math. Commun., 5 (2011), 69. doi: 10.3934/amc.2011.5.69. Google Scholar [11] P. Erdős, R. Graham, I. Z. Ruzsa and H. Taylor, Bounds for arrays of dots with distinct slopes or lengths,, Combinatorica, 12 (1992), 39. doi: 10.1007/BF01191203. Google Scholar [12] P. Erdős and P. Turán, On a problem of Sidon in additive number theory and on some related problems,, J. London Math. Soc., 16 (1941), 212. Google Scholar [13] T. Etzion, Combinatorial designs derived from Costas arrays,, in Sequences: Combinatorics, (1990), 208. Google Scholar [14] T. Etzion, Sequence folding, lattice tiling, and multidimensional coding,, IEEE Trans. Inform. Theory, 57 (2011), 4383. doi: 10.1109/TIT.2011.2146010. Google Scholar [15] A. Freedman and N. Levanon, Any two $N \times N$ Costas arrays must have at least one common ambiguity sidelobe if $N>3$ - a proof,, Proc. IEEE, 73 (1985), 1530. Google Scholar [16] E. N. Gilbert, Latin squares which contain no repeated digrams,, SIAM Review, 7 (1965), 189. doi: 10.1137/1007035. Google Scholar [17] S. Golomb, Algebraic constructions for Costas arrays,, J. Combin. Theory Ser. A, 37 (1984), 13. doi: 10.1016/0097-3165(84)90015-3. Google Scholar [18] S. W. Golomb and H. Taylor, Constructions and properties of Costas arrays,, Proc. IEEE, 72 (1984), 1143. doi: 10.1109/PROC.1984.12994. Google Scholar [19] S. Rickard, Database of Costas arrays,, accessed 2012., (2012). Google Scholar [20] S. Rickard, Searching for Costas arrays using periodicity properties,, in Proc. 2004 IMA Int. Conf. Math. Signal Proc., (2004). Google Scholar [21] J. C. Russo, K. G. Erickson and J. K. Beard, Costas array search technique that maximizes backtrack and symmetry exploitation,, in 44th Annual Conf. Inform. Sci. Systems, (2010). doi: 10.1109/CISS.2010.5464772. Google Scholar [22] H. Taylor, Non-attacking rooks with distinct differences,, Technical Report CSI-84-03-02, (1984), 84. Google Scholar [23] K. Taylor, S. Rickard and K. Drakakis, On Costas arrays with various types of symmetry,, in 16th Int. Conf. Digital Signal Proc., (2009). doi: 10.1109/ICDSP.2009.5201135. Google Scholar [24] J. L. Wodlinger, Costas Arrays, Golomb Rulers and Wavelength Isolation Sequence Pairs,, Master's thesis, (2012). Google Scholar
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