Symplectic spreads, planar functions, and mutually unbiased bases

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• 作者：Kanat Abdukhalikov
• 关键词：Mutually unbiased bases ; Symplectic spreads ; Finite semifields ; Orthogonal decompositions of Lie algebras ; Planar functions ; Pseudo ; planar functions ; Automorphism groups ; 05B25 ; 51A40 ; 51E15 ; 12K10 ; 20B25 ; 17B20 ; 05B20
• 刊名：Journal of Algebraic Combinatorics
• 出版年：2015
• 出版时间：June 2015
• 年：2015
• 卷：41
• 期：4
• 页码：1055-1077
• 全文大小：528 KB
• 参考文献：1.Abdukhalikov, K.S.: On a group of automorphisms of an orthogonal decomposition of a Lie algebra of type \(A_{2^m-1}\) , (Russian) Izv. Vyssh. Uchebn. Zaved. Mat. 10, 11鈥?4 (1991); English translation in Soviet Math. (Iz. VUZ) 35, 9鈥?2 (1991)
  2.Abdukhalikov, K.S.: Invariant integral lattices in Lie algebras of type \(A_{p^m-1}\) (Russian). Mat. Sb. 184(4), 61鈥?04 (1993); English translation in Russian Acad. Sci. Sb. Math. 78(2), 447鈥?78 (1994)
  3.Abdukhalikov, K.S.: Affine invariant and cyclic codes over \(p\) -adic numbers and finite rings. Des. Codes Cryptogr. 23, 343鈥?70 (2001)View Article MATH MathSciNet
  4.Abdukhalikov, K., Bannai, E., Suda, S.: Association schemes related to universally optimal configurations, Kerdock codes and extremal Euclidean line-sets. J. Comb. Theory Ser. A 116, 434鈥?48 (2009)View Article MATH MathSciNet
  5.Alltop, W.O.: Complex sequences with low periodic correlations. IEEE Trans. Inf. Theory 26(3), 350鈥?54 (1980)View Article MATH MathSciNet
  6.Bader, L., Kantor, W.M., Lunardon, G.: Symplectic spreads from twisted fields. Boll. Un. Math. Ital. A 7(8), 383鈥?89 (1994)MathSciNet
  7.Ball, S., Bamberg, J., Lavrauw, M., Penttila, T.: Symplectic spreads. Des. Codes Cryptogr. 32, 9鈥?4 (2004)View Article MATH MathSciNet
  8.Bandyopadhyay, S., Boykin, P.O., Roychowdhury, V., Vatan, F.: A new proof for the existence of mutually unbiased bases. Algorithmica 34, 512鈥?28 (2002)View Article MATH MathSciNet
  9.Benneth, C.H., Brassard, G.: Quantum cryptography: public key distribution and coin tossing. In: Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, 175, 1984
  10.Bierbrauer, J.: New semifields, PN and APN functions. Des. Codes Cryptogr. 54(3), 189鈥?00 (2010)View Article MATH MathSciNet
  11.Boykin, P.O., Sitharam, M., Tiep, P.H., Wocjan, P.: Mutually unbiased bases and orthogonal decompositions of Lie algebras. Quantum Inf. Comput. 7(4), 371鈥?82 (2007)MATH MathSciNet
  12.Budaghyan, L., Helleseth, T.: New perfect nonlinear multinomials over \(F_{p^{2k}}\) for any odd prime \(p\) . In: SETA 08: Proceedings of the 5th International Conference on Sequences and their Applications, 403鈥?14. Springer, Berlin (2008)
  13.Calderbank, A.R., Cameron, P.J., Kantor, W.M., Seidel, J.J.: \(\mathbb{Z}_4\) -Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets. Proc. Lond. Math. Soc. 3(75), 436鈥?80 (1997)View Article MathSciNet
  14.Cohen, S.D., Ganley, M.J.: Commutative semifields, two-dimensional over their middle nuclei. J. Algebra 75, 373鈥?85 (1982)View Article MATH MathSciNet
  15.Coulter, R.S., Matthews, R.W.: Planar functions and planes of Lenz鈥揃arlotti class II. Des. Codes Cryptogr. 10, 167鈥?84 (1997)View Article MATH MathSciNet
  16.Coulter, R.S., Henderson, M., Kosick, P.: Planar polynomials for commutative semifields with specified nuclei. Des. Codes Cryptogr. 44, 275鈥?86 (2007)View Article MATH MathSciNet
  17.Coulter, R.S., Kosick, P.: Commutative semifields of order 243 and 3125. In: Finite Fields: Theory and Applications. In: Contemp. Math., vol. 518, Am. Math. Soc., Providence, RI, 129鈥?36 (2010)
  18.Dembowski, P.: Finite geometries. Springer, Berlin (1968)
  19.Dickson, L.E.: On commutative linear algebras in which division is always uniquely possible. Trans. Am. Math. Soc. 7, 514鈥?22 (1906)View Article MATH
  20.Ding, C., Yuan, J.: A family of skew Hadamard difference sets. J. Combin. Theory Ser. A 113, 1526鈥?535 (2006)View Article MATH MathSciNet
  21.Ding, C., Yin, J.: Signal sets from functions with optimum nonlinearity. IEEE Trans. Commun. 55(5), 936鈥?40 (2007)View Article
  22.Ganley, M.J.: Central weak nucleus semifields. Eur. J. Combin. 2, 339鈥?47 (1981)View Article MATH MathSciNet
  23.Godsil, C., Roy, A.: Equiangular lines, mutually unbiased bases, and spin models. Eur. J. Combin. 30, 246鈥?62 (2009)View Article MATH MathSciNet
  24.Gow, R.: Generation of mutually unbiased bases as powers of a unitary matrix in 2-power dimensions. arXiv:鈥媘ath/鈥?703333
  25.Hammons, R., Kumar, P.V., Calderbank, A.R., Sloane, N.J.A., Sol茅, P.: The \(Z_4\) -linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inf. Theory 40(2), 301鈥?19 (1994)View Article MATH
  26.Ivanovi膰, I.D.: Geometrical description of quantal state determination. J. Phys. A 14, 3241鈥?245 (1981)View Article MathSciNet
  27.Kantor, W.M., Williams, M.E.: New flag-transitive affine planes of even order. J. Comb. Theory A 74, 1鈥?3 (1996)View Article MATH MathSciNet
  28.Kantor, W.M.: Ovoids and translation planes. Can. J. Math. 34, 1195鈥?207 (1982)View Article MATH MathSciNet
  29.Kantor, W.M.: Projective planes of order \(q\) whose collineation groups have order \(q^2\) . J. Algebr. Comb. 3, 405鈥?25 (1994)View Article MATH MathSciNet
  30.Kantor, W.M.: Note on Lie algebras, finite groups and finite geometries. In: Arasu, K.T., et al. (eds.) Groups, Difference Sets, and the Monster, pp. 73鈥?1. de Gruyter, Berlin (1996)
  31.Kantor, W.M.: Commutative semifields and symplectic spreads. J. Algebra 270, 96鈥?14 (2003)View Article MATH MathSciNet
  32.Kantor, W.M., Williams, M.E.: Symplectic semifield planes and \(\mathbb{Z}_4\) -linear codes. Trans. Am. Math. Soc. 356, 895鈥?38 (2004)View Article MATH MathSciNet
  33.Kantor, W.M., Williams, M.E.: Nearly flag-transitive affine planes. Adv. Geom. 10, 161鈥?83 (2010)View Article MATH MathSciNet
  34.Kantor, W.M.: MUBs inequivalence and affine planes. J. Math. Phys. 53, 032204 (2012)View Article MathSciNet
  35.Klappenecker, A., R枚tteler, M.: Constructions of mutually unbiased bases. Finite Fields and Applications. Lecture Notes in Computer Sciences, vol. 2948, pp. 137鈥?44. Springer, Berlin (2004)
  36.Knuth, D.E.: Finite semifields and projective planes. J. Algebra 2, 182鈥?17 (1965)View Article MATH MathSciNet
  37.Kostrikin, A.I., Tiep, P.H.: Orthogonal Decompositions and Integral Lattices. Walter de Gruyter, Berlin (1994)View Article MATH
  38.LeCompte, N., Martin, W.J., Owens, W.: On the equivalence between real mutually unbiased bases and a certain class of association schemes. Eur. J. Comb. 31, 1499鈥?512 (2010)View Article MATH MathSciNet
  39.Nechaev, A.A.: Kerdock code in a cyclic form (Russian). Diskret. Mat. 1(4), 123鈥?39 (1989); English translation in Discrete Math. Appl. 1(4), 365鈥?84 (1991)
  40.Penttila, T., Williams, B.: Ovoids of parabolic spaces. Geom. Dedicata 82, 1鈥?9 (2000)View Article MATH MathSciNet
  41.Roy, A., Scott, A.J.: Weighted complex projective 2-designs from bases: optimal state determination by orthogonal measurements. J. Math. Phys. 48, 072110 (2007)View Article MathSciNet
  42.Schmidt, K.-U., Zhou, Y.: Planar functions over fields of characteristic two. J. Algebr. Combin. 40, 503鈥?26 (2014)View Article MATH MathSciNet
  43.Schwinger, J.: Unitary operator bases. Proc. Natl. Acad. Sci. USA 46, 570鈥?79 (1960)View Article MATH MathSciNet
  44.Suetake, C.: A new class of translation planes of order \(q^3\) . Osaka J. Math. 22, 773鈥?86 (1985)MATH MathSciNet
  45.Thas, J.A., Payne, S.E.: Spreads and ovoids in finite generalized quadrangles. Geom. Dedicata 52, 227鈥?53 (1994)View Article MATH MathSciNet
  46.Tits, J.: Ovo茂des et groupes de Suzuki. Arch. Math. 13, 187鈥?98 (1962)View Article MATH MathSciNet
  47.Weng, G., Zeng, X.: Further results on planar DO functions and commutative semifields. Des. Codes Cryptogr. 63, 413鈥?23 (2012)View Article MATH MathSciNet
  48.Wootters, W.K., Fields, B.D.: Optimal state-determination by mutually unbiased measurements. Ann. Phys. 191(2), 363鈥?81 (1989)View Article MathSciNet
  49.Zha, Z., Kyureghyan, G.M., Wang, X.: Perfect nonlinear binomials and their semifields. Finite Fields Their Appl. 15, 125鈥?33 (2009)View Article MATH MathSciNet
  50.Zha, Z., Wang, X.: New families of perfect nonlinear polynomial functions. J. Algebra 322, 3912鈥?918 (2009)View Article MATH MathSciNet
  51.Zhou, Y.: \((2^n,2^n,2^n,1)\) -relative difference sets and their representations. J. Combin. Des. 21(12), 563鈥?84 (2013)MATH MathSciNet
  52.Zhou, Y., Pott, A.: A new family of semifields with 2 parameters. Adv. Math. 234, 43鈥?0 (2013)View Article MATH MathSciNet
  
• 作者单位：Kanat Abdukhalikov (1) (2)
  
  1. Department of Mathematical Sciences, UAE University, PO Box 15551, Al Ain, UAE
  2. Institute of Mathematics, Pushkin Str 125, Almaty, 050010, Kazakhstan
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Combinatorics
  Convex and Discrete Geometry
  Order, Lattices and Ordered Algebraic Structures
  Computer Science, general
  Group Theory and Generalizations
  
• 出版者：Springer U.S.
• ISSN：1572-9192

文摘

In this paper we give explicit descriptions of complete sets of mutually unbiased bases (MUBs) and orthogonal decompositions of special Lie algebras \(sl_n(\mathbb {C})\) obtained from commutative and symplectic semifields, and from some other non-semifield symplectic spreads. Relations between various constructions are also studied. We show that the automorphism group of a complete set of MUBs is isomorphic to the automorphism group of the corresponding orthogonal decomposition of the Lie algebra \(sl_n(\mathbb {C})\). In the case of symplectic spreads this automorphism group is determined by the automorphism group of the spread. By using the new notion of pseudo-planar functions over fields of characteristic two we give new explicit constructions of complete sets of MUBs.