Circulant matrices with orthogonal rows and off-diagonal entries of absolute value 1

Daniel Uzcátegui Contreras; Dardo Goyeneche; Ondřej Turek; Zuzana Václavíková

Communications in Mathematics (2021)

  • Issue: 1, page 15-34
  • ISSN: 1804-1388

Abstract

top
It is known that a real symmetric circulant matrix with diagonal entries d ≥ 0 , off-diagonal entries ± 1 and orthogonal rows exists only of order 2 d + 2 (and trivially of order 1 ) [Turek and Goyeneche 2019]. In this paper we consider a complex Hermitian analogy of those matrices. That is, we study the existence and construction of Hermitian circulant matrices having orthogonal rows, diagonal entries d ≥ 0 and any complex entries of absolute value 1 off the diagonal. As a particular case, we consider matrices whose off-diagonal entries are 4th roots of unity; we prove that the order of any such matrix with d different from an odd integer is n = 2 d + 2 . We also discuss a similar problem for symmetric circulant matrices defined over finite rings ℤ m . As an application of our results, we show a close connection to mutually unbiased bases, an important open problem in quantum information theory.

How to cite

top

Contreras, Daniel Uzcátegui, et al. "Circulant matrices with orthogonal rows and off-diagonal entries of absolute value $1$." Communications in Mathematics (2021): 15-34. <http://eudml.org/doc/297928>.

@article{Contreras2021,
abstract = {It is known that a real symmetric circulant matrix with diagonal entries $d\ge 0$, off-diagonal entries $\pm 1$ and orthogonal rows exists only of order $2d+2$ (and trivially of order $1$) [Turek and Goyeneche 2019]. In this paper we consider a complex Hermitian analogy of those matrices. That is, we study the existence and construction of Hermitian circulant matrices having orthogonal rows, diagonal entries $d\ge 0$ and any complex entries of absolute value $1$ off the diagonal. As a particular case, we consider matrices whose off-diagonal entries are 4th roots of unity; we prove that the order of any such matrix with $d$ different from an odd integer is $n=2d+2$. We also discuss a similar problem for symmetric circulant matrices defined over finite rings $\mathbb \{Z\}_m$. As an application of our results, we show a close connection to mutually unbiased bases, an important open problem in quantum information theory.},
author = {Contreras, Daniel Uzcátegui, Goyeneche, Dardo, Turek, Ondřej, Václavíková, Zuzana},
journal = {Communications in Mathematics},
keywords = {Circulant matrix; orthogonal matrix; Hadamard matrix; mutually unbiased base},
language = {eng},
number = {1},
pages = {15-34},
publisher = {University of Ostrava},
title = {Circulant matrices with orthogonal rows and off-diagonal entries of absolute value $1$},
url = {http://eudml.org/doc/297928},
year = {2021},
}

TY - JOUR
AU - Contreras, Daniel Uzcátegui
AU - Goyeneche, Dardo
AU - Turek, Ondřej
AU - Václavíková, Zuzana
TI - Circulant matrices with orthogonal rows and off-diagonal entries of absolute value $1$
JO - Communications in Mathematics
PY - 2021
PB - University of Ostrava
IS - 1
SP - 15
EP - 34
AB - It is known that a real symmetric circulant matrix with diagonal entries $d\ge 0$, off-diagonal entries $\pm 1$ and orthogonal rows exists only of order $2d+2$ (and trivially of order $1$) [Turek and Goyeneche 2019]. In this paper we consider a complex Hermitian analogy of those matrices. That is, we study the existence and construction of Hermitian circulant matrices having orthogonal rows, diagonal entries $d\ge 0$ and any complex entries of absolute value $1$ off the diagonal. As a particular case, we consider matrices whose off-diagonal entries are 4th roots of unity; we prove that the order of any such matrix with $d$ different from an odd integer is $n=2d+2$. We also discuss a similar problem for symmetric circulant matrices defined over finite rings $\mathbb {Z}_m$. As an application of our results, we show a close connection to mutually unbiased bases, an important open problem in quantum information theory.
LA - eng
KW - Circulant matrix; orthogonal matrix; Hadamard matrix; mutually unbiased base
UR - http://eudml.org/doc/297928
ER -

References

top
  1. Backelin, J., Square multiples n give infinitely many cyclic n -roots, 1989, Stockholms Universitet, Matematiska Institutionen, (1989) 
  2. Bandyopadhyay, S., Boykin, P.O., Roychowdhury, V., Vatan, and F., 10.1007/s00453-002-0980-7, Algorithmica, 34, 4, 2002, 512-528, Springer, (2002) MR1943521DOI10.1007/s00453-002-0980-7
  3. Tirkel, S.T. Blake and A.Z., A construction for perfect periodic autocorrelation sequences, International Conference on Sequences and Their Applications, 2014, 104-108, Springer, (2014) MR3297324
  4. Chu, D., 10.1109/TIT.1972.1054840, IEEE Transactions on information theory, 18, 4, 1972, 531-532, IEEE, (1972) DOI10.1109/TIT.1972.1054840
  5. Craigen, R., Kharaghani, H., On the nonexistence of Hermitian circulant complex Hadamard matrices, Australasian Journal of Combi natorics, 7, 1993, 225-228, (1993) MR1211281
  6. Craigen, R., 10.4153/CMB-1994-067-1, Canadian Mathematical Bulletin, 37, 4, 1994, 461-467, Cambridge University Press, (1994) MR1303672DOI10.4153/CMB-1994-067-1
  7. Faugère, J.-C., Finding all the solutions of Cyclic 9 using Gr{ö}bner basis techniques, Lecture Notes Series on Computing -- Computer Mathematics: Proceedings of the Fifth Asian Symposium (ASCM 2001), 9, 2001, 1-12, World Scientific, (2001) MR1877437
  8. Farnett, E.C., Stevens, G.H., Pulse Compression Radar, Radar Handbook, 2nd edition, 1990, 10.1-10.39, McGraw-Hill, New York, (1990) 
  9. Hiranandani, G., Schlenker, J.-M., 10.1016/j.ejc.2015.05.010, European Journal of Combinatorics, 51, 2016, 306-314, Elsevier, (2016) MR3398859DOI10.1016/j.ejc.2015.05.010
  10. Heimiller, R., 10.1109/TIT.1961.1057655, IRE Transactions on Information Theory, 7, 4, 1961, 254-257, IEEE, (1961) DOI10.1109/TIT.1961.1057655
  11. Ivonovic, I.D., 10.1088/0305-4470/14/12/019, Journal of Physics A: Mathematical and General, 14, 12, 1981, 3241-3245, IOP Publishing, (1981) MR0639558DOI10.1088/0305-4470/14/12/019
  12. Ipatov, V.P., Spread Spectrum and CDMA: Principles and Applications, 2005, John Wiley & Sons, (2005) 
  13. Liu, Y., Fan, P., 10.1049/el:20040437, Electronics Letters, 40, 10, 2004, 598-599, IET, (2004) DOI10.1049/el:20040437
  14. Milewski, A., 10.1147/rd.275.0426, IBM Journal of Research and Development, 27, 5, 1983, 426-431, IBM, (1983) DOI10.1147/rd.275.0426
  15. Mow, W.H., A study of correlation of sequences, 1993, PhD Thesis, Department of Information Engineering, The Chinese University of Hong Kong. (1993) 
  16. Ryser, H.J., Combinatorial mathematics, The Carus Mathematical Monographs, 14, 1963, The Mathematical Association of America, John Wiley and Sons, Inc., New York, (1963) MR0150048
  17. Scott, A.J., 10.1088/0305-4470/39/43/009, Journal of Physics A: Mathematical and General, 39, 43, 2006, 13507, IOP Publishing, (2006) MR2269701DOI10.1088/0305-4470/39/43/009
  18. Turek, O., Goyeneche, D., 10.1016/j.laa.2019.01.018, Linear Algebra and its Applications, 569, 2019, 241-265, Elsevier, (2019) MR3905223DOI10.1016/j.laa.2019.01.018
  19. Wootters, W.K., Fields, B.D., 10.1016/0003-4916(89)90322-9, Annals of Physics, 191, 2, 1989, 363-381, Elsevier, (1989) MR1003014DOI10.1016/0003-4916(89)90322-9
  20. Xu, L., Phase coded waveform design for Sonar Sensor Network, Conference on Communications and Networking in China (CHINACOM), 2011 6th International ICST, 2011, 251-256, Springer, (2011) 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.