Geodesic
Circulant matrices with orthogonal rows and off-diagonal entries of absolute value $1$
Communications in Mathematics, Tome 29 (2021) no. 1, pp. 15-34 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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It is known that a real symmetric circulant matrix with diagonal entries $d\geq 0$, 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 $d\geq 0$ 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.
It is known that a real symmetric circulant matrix with diagonal entries $d\geq 0$, 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 $d\geq 0$ 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.
Classification : 15B05, 15B10, 15B36
Keywords: Circulant matrix; orthogonal matrix; Hadamard matrix; mutually unbiased base 
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     title = {Circulant matrices with orthogonal rows and off-diagonal entries of absolute value $1$},
     journal = {Communications in Mathematics},
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Contreras, Daniel Uzcátegui; Goyeneche, Dardo; Turek, Ondřej; Václavíková, Zuzana. Circulant matrices with orthogonal rows and off-diagonal entries of absolute value $1$. Communications in Mathematics, Tome 29 (2021) no. 1, pp. 15-34. http://geodesic.mathdoc.fr/item/COMIM_2021_29_1_a1/

[1] Backelin, J.: Square multiples $n$ give infinitely many cyclic $n$-roots. 1989, Stockholms Universitet, Matematiska Institutionen,

[2] Bandyopadhyay, S., Boykin, P.O., Roychowdhury, V., Vatan, and F.: A new proof for the existence of mutually unbiased bases. Algorithmica, 34, 4, 2002, 512-528, Springer, | DOI | MR

[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, | MR

[4] Chu, D.: Polyphase Codes With Good Periodic Correlation Properties. IEEE Transactions on information theory, 18, 4, 1972, 531-532, IEEE, | DOI

[5] Craigen, R., Kharaghani, H.: On the nonexistence of Hermitian circulant complex Hadamard matrices. Australasian Journal of Combi natorics, 7, 1993, 225-228, | MR

[6] Craigen, R.: Trace, symmetry and orthogonality. Canadian Mathematical Bulletin, 37, 4, 1994, 461-467, Cambridge University Press, | DOI | MR

[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, | MR

[8] Farnett, E.C., Stevens, G.H.: Pulse Compression Radar. Radar Handbook, 2nd edition, 1990, 10.1-10.39, McGraw-Hill, New York,

[9] Hiranandani, G., Schlenker, J.-M.: Small circulant complex Hadamard matrices of Butson type. European Journal of Combinatorics, 51, 2016, 306-314, Elsevier, | DOI | MR

[10] Heimiller, R.: Phase shift pulse codes with good periodic correlation properties. IRE Transactions on Information Theory, 7, 4, 1961, 254-257, IEEE, | DOI

[11] Ivonovic, I.D.: Geometrical description of quantal state determination. Journal of Physics A: Mathematical and General, 14, 12, 1981, 3241-3245, IOP Publishing, | DOI | MR

[12] Ipatov, V.P.: Spread Spectrum and CDMA: Principles and Applications. 2005, John Wiley & Sons,

[13] Liu, Y., Fan, P.: Modified Chu sequences with smaller alphabet size. Electronics Letters, 40, 10, 2004, 598-599, IET, | DOI

[14] Milewski, A.: Periodic sequences with optimal properties for channel estimation and fast start-up equalization. IBM Journal of Research and Development, 27, 5, 1983, 426-431, IBM, | DOI

[15] Mow, W.H.: A study of correlation of sequences. 1993, PhD Thesis, Department of Information Engineering, The Chinese University of Hong Kong.

[16] Ryser, H.J.: Combinatorial mathematics. The Carus Mathematical Monographs, 14, 1963, The Mathematical Association of America, John Wiley and Sons, Inc., New York, | MR

[17] Scott, A.J.: Tight informationally complete quantum measurements. Journal of Physics A: Mathematical and General, 39, 43, 2006, 13507, IOP Publishing, | DOI | MR

[18] Turek, O., Goyeneche, D.: A generalization of circulant Hadamard and conference matrices. Linear Algebra and its Applications, 569, 2019, 241-265, Elsevier, | DOI | MR

[19] Wootters, W.K., Fields, B.D.: Optimal state-determination by mutually unbiased measurements. Annals of Physics, 191, 2, 1989, 363-381, Elsevier, | DOI | MR

[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,