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On the measurement of the (non)linearity of Costas permutations. (English) Zbl 1189.05006

Summary: We study several criteria for the (non)linearity of Costas permutations, with or without the imposition of additional algebraic structure in the domain and the range of the permutation, aiming to find one that successfully identifies Costas permutations as more nonlinear than randomly chosen permutations of the same order.

MSC:

05A05 Permutations, words, matrices
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References:

[1] J. P. Costas, “Medium constraints on sonar design and performance,” Tech. Rep. R65EMH33, GE Co., 1965.
[2] J. P. Costas, “A study of detection waveforms having nearly ideal range-doppler ambiguity properties,” Proceedings of the IEEE, vol. 72, no. 8, pp. 996-1009, 1984. · doi:10.1109/PROC.1984.12967
[3] K. Drakakis, “A review of Costas arrays,” Journal of Applied Mathematics, vol. 2006, Article ID 26385, 32 pages, 2006. · Zbl 1188.94028 · doi:10.1155/JAM/2006/26385
[4] S. W. Golomb, “Algebraic constructions for Costas arrays,” Journal of Combinatorial Theory Series A, vol. 37, no. 1, pp. 13-21, 1984. · Zbl 0547.05020 · doi:10.1016/0097-3165(84)90015-3
[5] S. W. Golomb and H. Taylor, “Constructions and properties of Costas arrays,” Proceedings of the IEEE, vol. 72, no. 9, pp. 1143-1163, 1984. · Zbl 1200.05043 · doi:10.1109/PROC.1984.12994
[6] K. Drakakis, R. Gow, and G. McGuire, “APN permutations on \Bbb Zn and Costas arrays,” Discrete Applied Mathematics, vol. 157, no. 15, pp. 3320-3326, 2009. · Zbl 1227.05014 · doi:10.1016/j.dam.2009.06.029
[7] C. Carlet and C. Ding, “Highly nonlinear mappings,” Journal of Complexity, vol. 20, no. 2-3, pp. 205-244, 2004. · Zbl 1053.94011 · doi:10.1016/j.jco.2003.08.008
[8] A. Pott, “Nonlinear functions in abelian groups and relative difference sets,” Discrete Applied Mathematics, vol. 138, no. 1-2, pp. 177-193, 2004. · Zbl 1035.05023 · doi:10.1016/S0166-218X(03)00293-2
[9] K. Drakakis, V. Requena, and G. McGuire, “On the nonlinearity of exponential welch costas functions,” IEEE Transactions on Information Theory, vol. 56, no. 3, pp. 1230-1238, 2010. · Zbl 1366.94486 · doi:10.1109/TIT.2009.2039164
[10] K. Drakakis, R. Gow, and L. O’Carroll, “On the symmetry of Welch- and Golomb-constructed Costas arrays,” Discrete Mathematics, vol. 309, no. 8, pp. 2559-2563, 2009. · Zbl 1193.05041 · doi:10.1016/j.disc.2008.04.058
[11] K. Drakakis, S. Rickard, J. K. Beard, et al., “Results of the enumeration of Costas arrays of order 27,” IEEE Transactions on Information Theory, vol. 54, no. 10, pp. 4684-4687, 2008. · Zbl 1322.05008 · doi:10.1109/TIT.2008.928979
[12] K. Drakakis, “Data mining and costas arrays,” Turkish Journal of Electrical Engineering and Computer Sciences, vol. 15, no. 1, pp. 67-76, 2007.
[13] K. Drakakis, “Three challenges in Costas arrays,” Ars Combinatoria, vol. 89, pp. 167-182, 2008. · Zbl 1224.05063
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