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## Open Mathematics

2014 | 12 | 2 | 337-348
Tytuł artykułu

### Construction of the mutually orthogonal extraordinary supersquares

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Our purpose is to determine the complete set of mutually orthogonal squares of order d, which are not necessary Latin. In this article, we introduce the concept of supersquare of order d, which is defined with the help of its generating subgroup in $$\mathbb{F}_d \times \mathbb{F}_d$$. We present a method of construction of the mutually orthogonal supersquares. Further, we investigate the orthogonality of extraordinary supersquares, a special family of squares, whose generating subgroups are extraordinary. The extraordinary subgroups in $$\mathbb{F}_d \times \mathbb{F}_d$$ are of great importance in the field of quantum information processing, especially for the study of mutually unbiased bases. We determine the most general complete sets of mutually orthogonal extraordinary supersquares of order 4, which consist in the so-called Type I and Type II. The well-known case of d − 1 mutually orthogonal Latin squares is only a special case, namely Type I.
Słowa kluczowe
EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
337-348
Opis fizyczny
Daty
wydano
2014-02-01
online
2013-11-21
Twórcy
autor
• University Politehnica of Bucharest
autor
• University of Bucharest
Bibliografia
• [1] Aly Ahmed S.A.A., Quantum Error Control Codes, PhD thesis, Texas A&M University, 2008
• [2] Asplund J., Keranen M.S., Mutually orthogonal equitable Latin rectangles, Discrete Math., 2011, 311(12), 1015–1033 http://dx.doi.org/10.1016/j.disc.2011.03.003
• [3] Bandyopadhyay S., Boykin P.O., Roychowdhury V., Vatan V., A new proof for the existence of mutually unbiased bases, Algorithmica, 2002, 34(4), 512–528 http://dx.doi.org/10.1007/s00453-002-0980-7
• [4] Bose R.C., Shrikhande S.S., On the construction of sets of mutually orthogonal Latin squares and the falsity of a conjecture of Euler, Trans. Amer. Math. Soc., 1960, 95(2), 191–209 http://dx.doi.org/10.1090/S0002-9947-1960-0111695-3
• [5] Ghiu I., A new method of construction of all sets of mutually unbiased bases for two-qubit systems, J. Phys. Conf. Ser., 2012, 338, #012008
• [6] Ghiu I., Generation of all sets of mutually unbiased bases for three-qubit systems, Phys. Scr., 2013, T153, #014027
• [7] Hall J.L., Rao A., Mutually orthogonal Latin squares from the inner products of vectors in mutually unbiased bases, J. Phys. A, 2010, 43(13), #135302 http://dx.doi.org/10.1088/1751-8113/43/13/135302
• [8] Hayashi A., Horibe M., Hashimoto T., Mean king’s problem with mutually unbiased bases and orthogonal Latin squares, Phys. Rev. A, 2005, 71(5), #052331
• [9] Huczynska S., Mullen G.L., Frequency permutation arrays, J. Combin. Des., 2006, 14(6), 463–478 http://dx.doi.org/10.1002/jcd.20096
• [10] Keedwell A.D., Mullen G.L., Sets of partially orthogonal Latin squares and projective planes, Discrete Math., 2004, 288(1–3), 49–60 http://dx.doi.org/10.1016/j.disc.2004.04.014
• [11] Khanban A.A., Mahdian M., Mahmoodian E.S., A linear algebraic approach to orthogonal arrays and Latin squares, Ars Combin., 2012, 105, 3–13
• [12] Kim K., Prasanna Kumar V.K., Perfect Latin squares and parallel array access, In: The 16th Annual International Symposium on Computer Architecture, Jerusalem, May 28–June 1, 1989, IEEE Computer Society Press, Washington-Los Alamitos-Brussels-Tokyo, 1989, 372–379
• [13] Klimov A.B., Romero J.L., Björk G., Sánchez-Soto L.L., Geometrical approach to mutually unbiased bases, J. Phys. A, 2007, 40(14), 3987–3998 http://dx.doi.org/10.1088/1751-8113/40/14/014
• [14] Laywine C.F., Mullen G.L., Generalizations of Bose’s equivalence between complete sets of mutually orthogonal Latin squares and affine planes, J. Combin. Theory Ser. A, 1992, 61(1), 13–35 http://dx.doi.org/10.1016/0097-3165(92)90050-5
• [15] Laywine C.F., Mullen G.L., Discrete Mathematics Using Latin Squares, Wiley-Intersci. Ser. Discrete Math. Optim., John Wiley & Sons, 1998
• [16] Paterek T., Dakic B., Brukner Č., Mutually unbiased bases, orthogonal Latin squares, and hidden-variable models, Phys. Rev. A, 2009, 79(1), #012109
• [17] Paterek T., Pawłowski M., Grassl M., Brukner Č., On the connection between mutually unbiased bases and orthogonal Latin squares, Phys. Scr., 2010, T140, #014031
Typ dokumentu
Bibliografia
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