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MORE CONSTRUCTIONS OF APPROXIMATELY MUTUALLY UNBIASED BASES

Part of: Finite fields and commutative rings (number-theoretic aspects)

Published online by Cambridge University Press:  17 August 2015

XIWANG CAO  and
WUN-SENG CHOU
XIWANG CAO*
Affiliation:
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing 100093, PR China email [email protected]
WUN-SENG CHOU
Affiliation:
Institute of Mathematics, Academia Sinica, Taiwan Department of Mathematical Sciences, National Chengchi University, Taipei, Taiwan email [email protected]
*
✉[email protected]
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Abstract

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Let $m$ be a positive integer and $p$ a prime number. We prove the orthogonality of some character sums over the finite field $\mathbb{F}_{p^{m}}$ or over a subset of a finite field and use this to construct some new approximately mutually unbiased bases of dimension $p^{m}$ over the complex number field $\mathbb{C}$, especially with $p=2$.

Keywords

finite fieldcharacter sumapproximately mutually unbiased basesquantum information theory

MSC classification

Primary: 11T71: Algebraic coding theory; cryptography
Secondary: 81P70: Quantum coding (general)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 93 , Issue 2 , April 2016 , pp. 211 - 222
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Bandyopadhyay, S., Boykin, P. O., Roychowdhury, V. and Vatan, F., ‘A new proof of the existence of mutually unbiased bases’, Algorithmica 34 (2002), 512–528.Google Scholar
Dillon, J. and Dobbertin, H., ‘New cyclic difference sets with Singer parameters’, Finite Fields Appl. 10 (2004), 342–389.Google Scholar
Durt, T., Englert, B. G., Bengtsson, I. and Zyczkowski, K., ‘On mutually unbiased bases’, Int. J. Quantum Inf. 8 (2010), 535–640.Google Scholar
Godsil, C. and Roy, A., ‘Equiangular lines, mutually unbiased bases and spin models’, European J. Combin. 30 (2009), 246–262.CrossRefGoogle Scholar
Hall, J., ‘Mutually unbiased bases and related structures’, PhD Thesis, RMIT University, Melbourne, 2011.Google Scholar
Helleseth, T., ‘Some results about the cross-correlation function between two maximal linear sequences’, Discrete Math. 16 (1976), 209–232.CrossRefGoogle Scholar
Ivanovic, I. D., ‘Geometrical description of quantal state determination’, J. Phys. A 14 (1981), 3241–3245.Google Scholar
Johansen, A., Helleseth, T. and Kholosha, A., ‘Further results on m-sequences with five-valued cross correlation’, IEEE Trans. Inform. Theory 55 (2009), 5792–5802.Google Scholar
Klappenecker, A. and Rötteler, M., ‘Constructions of mutually unbiased bases’, Fq 7 2003, Lecture Notes in Computer Science, 2948 (eds. Mullen, G., Poli, A. and Stichtenoth, H.) (Springer, Berlin, 2003), 137–144.Google Scholar
Klappenecker, A. and Rötteler, M., ‘Mutually unbiased bases are complex projective 2-designs’, Proc. 2005 IEEE Intl Symp. on Information Theory, Adelaide, Australia (2005), 1740–1744.Google Scholar
Klappenecker, A., Rötteler, M. and Shparlinski, I. E., ‘On approximately symmetric informationally complete positive operator-valued measures and related systems of quantum states’, J. Math. Phys. 46 (2005), 082104.Google Scholar
Lachaud, G. and Wolfmann, J., ‘The weights of the orthogonals of the extended quadratic binary Goppa codes’, IEEE Trans. Inform. Theory 36 (1990), 686–692.CrossRefGoogle Scholar
Lahtonen, J., McGuire, G. and Ward, H. W., ‘Gold and Kasami-Welch functions, quadratic forms and bent functions’, Adv. Math. Commun. 1 (2007), 243–250.CrossRefGoogle Scholar
Lidl, R. and Niederriter, H., Finite Fields, Encyclopedia of Mathematics and its Applications, 20 (Addison-Wesley, Reading, MA, 1983).Google Scholar
Lisonêk, P., ‘On the connection between Kloosterman sums and elliptic curves’, SETA 2008, Lecture Notes in Computer Science, 5203 (eds. Golomb, S. W. et al. ) (Springer, Berlin, 2008), 182–187.Google Scholar
Moreno, C. J. and Moreno, O., ‘Exponential sums and Goppa codes: I’, Proc. Amer. Math. Soc. 111 (1991), 523–531.Google Scholar
Planat, M. and Rosu, H., ‘Mutually unbiased bases, phase uncertainties and Gauss sums’, Eur. Phys. J. D 36 (2005), 133–139.Google Scholar
Rao, A., Donovan, D. and Hall, J. L., ‘Mutually orthogonal Latin squares and mutually unbiased bases in dimensions of odd prime power’, Cryptogr. Commun. 2 (2010), 221–231.Google Scholar
Saniga, M. and Planat, M., ‘Hjelmslev geometry of mutually unbiased bases’, J. Phys. A 39 (2006), 435–440.Google Scholar
Schwinger, J., ‘Unitary operator bases’, Proc. Natl. Acad. Sci. USA 46 (1960), 570–579.CrossRefGoogle ScholarPubMed
Shanbhag, A. G., Kumar, P. V. and Helleseth, T., ‘An upper bound for the extended Kloosterman sums over Galois rings’, Finite Fields Appl. 4 (1998), 218–238.CrossRefGoogle Scholar
Shparlinski, I. E. and Winterhof, A., Construction of Approximately Mutually Unbiased Bases, Lecture Notes in Computer Science, 3887 (Springer, Berlin, 2006), 793–799.Google Scholar
Wang, W. Y., Zhang, A. X. and Feng, K. Q., ‘Constructions of approximately mutually unbiased bases and symmetric informationally complete positive operator-valued measures by Gauss and Jacobi sums’, Sci. Sin. Math. 42 (2012), 971–984; (in Chinese).Google Scholar
Xu, G., ‘Compressed sensing matrices from Fourier matrices’, IEEE Trans. Inform. Theory 61 (2014), 469–478.CrossRefGoogle Scholar