Analytic aspects of the circulant Hadamard conjecture

Teodor Banica; Ion Nechita; Jean-Marc Schlenker

doi:10.5802/ambp.334

Analytic aspects of the circulant Hadamard conjecture
[Aspects analytiques de la conjecture d’Hadamard circulante]

Teodor Banica ¹ ; Ion Nechita ² ; Jean-Marc Schlenker ³

¹ Department of Mathematics, Cergy-Pontoise University, 95000 Cergy-Pontoise, France
² CNRS, Laboratoire de Physique Théorique, IRSAMC, Université de Toulouse, UPS, 31062 Toulouse, France
³ University of Luxembourg, Campus Kirchberg, Mathematics Research Unit, 6 rue Richard Coudenhove-Kalergi, L-1359 Luxembourg

Annales mathématiques Blaise Pascal, Tome 21 (2014) no. 1, pp. 25-59.

Résumé
Abstract

On étudie la question de comptage pour les matrices d’Hadamard réelles ou complexes circulantes, en utilisant des méthodes analytiques. Notre remarque principale est que pour $| q_{0} | = ... = | q_{N - 1} | = 1$ la quantité $Φ = \sum_{i + k = j + l} \frac{q_{i} q_{k}}{q_{j} q_{l}}$ satisfait $Φ \geq N^{2}$ , avec égalité si et seulement si $q = (q_{i})$ est le vecteur des valeurs propres d’une matrice d’Hadamard complexe circulante. Ceci suggère trois problèmes analytiques, à savoir : (1) la minimisation directe de $Φ$ , (2) l’étude des points critiques de $Φ$ , et (3) le calcul des moments de $Φ$ . On explore ici ces questions, avec plusieurs résultats et conjectures.

We investigate the problem of counting the real or complex Hadamard matrices which are circulant, by using analytic methods. Our main observation is the fact that for $| q_{0} | = ... = | q_{N - 1} | = 1$ the quantity $Φ = \sum_{i + k = j + l} \frac{q_{i} q_{k}}{q_{j} q_{l}}$ satisfies $Φ \geq N^{2}$ , with equality if and only if $q = (q_{i})$ is the eigenvalue vector of a rescaled circulant complex Hadamard matrix. This suggests three analytic problems, namely: (1) the brute-force minimization of $Φ$ , (2) the study of the critical points of $Φ$ , and (3) the computation of the moments of $Φ$ . We explore here these questions, with some results and conjectures.

MR Zbl

DOI : 10.5802/ambp.334

Classification : 05B20
Keywords: Circulant Hadamard matrix
Mot clés : Matrice d’Hadamard circulante

Affiliations des auteurs :

Teodor Banica ¹ ; Ion Nechita ² ; Jean-Marc Schlenker ³

@article{AMBP_2014__21_1_25_0,
     author = {Teodor Banica and Ion Nechita and Jean-Marc Schlenker},
     title = {Analytic aspects of the circulant {Hadamard} conjecture},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {25--59},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {21},
     number = {1},
     year = {2014},
     doi = {10.5802/ambp.334},
     mrnumber = {3248220},
     zbl = {1297.05042},
     language = {en},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.334/}
}

TY  - JOUR
AU  - Teodor Banica
AU  - Ion Nechita
AU  - Jean-Marc Schlenker
TI  - Analytic aspects of the circulant Hadamard conjecture
JO  - Annales mathématiques Blaise Pascal
PY  - 2014
SP  - 25
EP  - 59
VL  - 21
IS  - 1
PB  - Annales mathématiques Blaise Pascal
UR  - https://ambp.centre-mersenne.org/articles/10.5802/ambp.334/
DO  - 10.5802/ambp.334
LA  - en
ID  - AMBP_2014__21_1_25_0
ER  -

%0 Journal Article
%A Teodor Banica
%A Ion Nechita
%A Jean-Marc Schlenker
%T Analytic aspects of the circulant Hadamard conjecture
%J Annales mathématiques Blaise Pascal
%D 2014
%P 25-59
%V 21
%N 1
%I Annales mathématiques Blaise Pascal
%U https://ambp.centre-mersenne.org/articles/10.5802/ambp.334/
%R 10.5802/ambp.334
%G en
%F AMBP_2014__21_1_25_0

Teodor Banica; Ion Nechita; Jean-Marc Schlenker. Analytic aspects of the circulant Hadamard conjecture. Annales mathématiques Blaise Pascal, Tome 21 (2014) no. 1, pp. 25-59. doi : 10.5802/ambp.334. https://ambp.centre-mersenne.org/articles/10.5802/ambp.334/

Bibliographie
Cité par

[1] S. S. Agaian Hadamard matrices and their applications, Lecture Notes in Mathematics, 1168, Springer-Verlag, Berlin, 1985, pp. iii+227 | MR | Zbl

[2] K. T. Arasu; Warwick de Launey; S. L. Ma On circulant complex Hadamard matrices, Des. Codes Cryptogr., Volume 25 (2002) no. 2, pp. 123-142 | DOI | MR | Zbl

[3] Jörgen Backelin Square multiples n give infinitely many cyclic n-roots, Reports/Univ. of Stockholm (1989)

[4] Teo Banica; Gaurush Hiranandani; Ion Nechita; Jean-Marc Schlenker Small circulant complex Hadamard matrices of Butson type, arXiv preprint arXiv:1311.5390 (2013)

[5] Teo Banica; Ion Nechita; Jean-Marc Schlenker Submatrices of Hadamard matrices: complementation results, arXiv preprint arXiv:1311.0764 (2013) | MR

[6] Teodor Banica The Gale-Berlekamp game for complex Hadamard matrices, arXiv preprint arXiv:1310.1810 (2013) | MR

[7] Teodor Banica; Benoît Collins; Jean-Marc Schlenker On orthogonal matrices maximizing the 1-norm, Indiana Univ. Math. J., Volume 59 (2010) no. 3, pp. 839-856 | DOI | MR | Zbl

[8] Teodor Banica; Benoit Collins; Jean-Marc Schlenker On polynomial integrals over the orthogonal group, J. Combin. Theory Ser. A, Volume 118 (2011) no. 3, pp. 778-795 | DOI | MR | Zbl

[9] Teodor Banica; Ion Nechita Almost Hadamard matrices: the case of arbitrary exponents, Discrete Appl. Math., Volume 161 (2013) no. 16-17, pp. 2367-2379 | DOI | MR | Zbl

[10] Teodor Banica; Ion Nechita; Karol Życzkowski Almost Hadamard matrices: general theory and examples, Open Syst. Inf. Dyn., Volume 19 (2012) no. 4, pp. 1250024, 26 | DOI | MR | Zbl

[11] Ingemar Bengtsson; Wojciech Bruzda; Åsa Ericsson; Jan-Åke Larsson; Wojciech Tadej; Karol Życzkowski Mutually unbiased bases and Hadamard matrices of order six, J. Math. Phys., Volume 48 (2007) no. 5, pp. 052106, 21 | DOI | MR | Zbl

[12] Göran Björck Functions of modulus $1$ on $Z_{n}$ whose Fourier transforms have constant modulus, and “cyclic $n$ -roots”, Recent advances in Fourier analysis and its applications (Il Ciocco, 1989) (NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.), Volume 315, Kluwer Acad. Publ., Dordrecht, 1990, pp. 131-140 | MR | Zbl

[13] Göran Björck; Ralf Fröberg A faster way to count the solutions of inhomogeneous systems of algebraic equations, with applications to cyclic $n$ -roots, J. Symbolic Comput., Volume 12 (1991) no. 3, pp. 329-336 | DOI | MR | Zbl

[14] Goran Bjorck; Uffe Haagerup All cyclic p-roots of index 3, found by symmetry-preserving calculations, arXiv preprint arXiv:0803.2506 (2008)

[15] A. T. Butson Generalized Hadamard matrices, Proc. Amer. Math. Soc., Volume 13 (1962), pp. 894-898 | DOI | MR | Zbl

[16] Benoît Collins; Piotr Śniady Integration with respect to the Haar measure on unitary, orthogonal and symplectic group, Comm. Math. Phys., Volume 264 (2006) no. 3, pp. 773-795 | DOI | MR | Zbl

[17] R. Craigen; H. Kharaghani On the nonexistence of Hermitian circulant complex Hadamard matrices, Australas. J. Combin., Volume 7 (1993), pp. 225-227 | MR | Zbl

[18] Jean-Charles Faugère Finding all the solutions of Cyclic $9$ using Gröbner basis techniques, Computer mathematics (Matsuyama, 2001) (Lecture Notes Ser. Comput.), Volume 9, World Sci. Publ., River Edge, NJ, 2001, pp. 1-12 | MR | Zbl

[19] John Gilbert; Ziemowit Rzeszotnik The norm of the Fourier transform on finite abelian groups, Ann. Inst. Fourier (Grenoble), Volume 60 (2010) no. 4, pp. 1317-1346 | DOI | Numdam | MR | Zbl

[20] T. Gorin Integrals of monomials over the orthogonal group, J. Math. Phys., Volume 43 (2002) no. 6, pp. 3342-3351 | DOI | MR | Zbl

[21] D. Goyeneche Mutually unbiased triplets from non-affine families of complex Hadamard matrices in dimension 6, J. Phys. A, Volume 46 (2013) no. 10, pp. 105301, 15 | DOI | MR | Zbl

[22] Uffe Haagerup Orthogonal maximal abelian $*$ -subalgebras of the $n \times n$ matrices and cyclic $n$ -roots, Operator algebras and quantum field theory (Rome, 1996), Int. Press, Cambridge, MA, 1997, pp. 296-322 | MR | Zbl

[23] Uffe Haagerup Cyclic p-roots of prime lengths p and related complex Hadamard matrices, arXiv preprint arXiv:0803.2629 (2008) | MR

[24] Pierre de la Harpe; Vaughan Jones Paires de sous-algèbres semi-simples et graphes fortement réguliers, C. R. Acad. Sci. Paris Sér. I Math., Volume 311 (1990) no. 3, pp. 147-150 | MR | Zbl

[25] K. J. Horadam Hadamard matrices and their applications, Princeton University Press, Princeton, NJ, 2007, pp. xiv+263 | MR | Zbl

[26] Jonathan Jedwab; Sheelagh Lloyd A note on the nonexistence of Barker sequences, Des. Codes Cryptogr., Volume 2 (1992) no. 1, pp. 93-97 | DOI | MR | Zbl

[27] V. Jones; V. S. Sunder Introduction to subfactors, London Mathematical Society Lecture Note Series, 234, Cambridge University Press, Cambridge, 1997, pp. xii+162 | DOI | MR | Zbl

[28] T. Y. Lam; K. H. Leung On vanishing sums of roots of unity, J. Algebra, Volume 224 (2000) no. 1, pp. 91-109 | DOI | MR | Zbl

[29] Warwick de Launey On the nonexistence of generalised weighing matrices, Ars Combin., Volume 17 (1984) no. A, pp. 117-132 | MR | Zbl

[30] Warwick de Launey; David A. Levin A Fourier-analytic approach to counting partial Hadamard matrices, Cryptogr. Commun., Volume 2 (2010) no. 2, pp. 307-334 | DOI | MR | Zbl

[31] Ka Hin Leung; Bernhard Schmidt New restrictions on possible orders of circulant Hadamard matrices, Des. Codes Cryptogr., Volume 64 (2012) no. 1-2, pp. 143-151 | DOI | MR | Zbl

[32] Georg Pólya Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz, Math. Ann., Volume 84 (1921) no. 1-2, pp. 149-160 | DOI | MR

[33] Sorin Popa Orthogonal pairs of $*$ -subalgebras in finite von Neumann algebras, J. Operator Theory, Volume 9 (1983) no. 2, pp. 253-268 | MR | Zbl

[34] T. Prosen; T. H. Seligman; H. A. Weidenmüller Integration over matrix spaces with unique invariant measures, J. Math. Phys., Volume 43 (2002) no. 10, pp. 5135-5144 | DOI | MR | Zbl

[35] Herbert John Ryser Combinatorial mathematics, The Carus Mathematical Monographs, No. 14, Published by The Mathematical Association of America, 1963, pp. xiv+154 | MR | Zbl

[36] Bernhard Schmidt Cyclotomic integers and finite geometry, J. Amer. Math. Soc., Volume 12 (1999) no. 4, pp. 929-952 | DOI | MR | Zbl

[37] Ferenc Szöllősi Exotic complex Hadamard matrices and their equivalence, Cryptogr. Commun., Volume 2 (2010) no. 2, pp. 187-198 | DOI | MR | Zbl

[38] Ferenc Szöllősi A two-parameter family of complex Hadamard matrices of order 6 induced by hypocycloids, Proc. Amer. Math. Soc., Volume 138 (2010) no. 3, pp. 921-928 | DOI | MR | Zbl

[39] Wojciech Tadej; Karol Życzkowski A concise guide to complex Hadamard matrices, Open Syst. Inf. Dyn., Volume 13 (2006) no. 2, pp. 133-177 | DOI | MR | Zbl

[40] Terence Tao Fuglede’s conjecture is false in 5 and higher dimensions, Math. Res. Lett., Volume 11 (2004) no. 2-3, pp. 251-258 | DOI | MR | Zbl

[41] Terence Tao An uncertainty principle for cyclic groups of prime order, Math. Res. Lett., Volume 12 (2005) no. 1, pp. 121-127 | DOI | MR | Zbl

[42] Richard J. Turyn Character sums and difference sets, Pacific J. Math., Volume 15 (1965), pp. 319-346 | DOI | MR | Zbl

[43] R. F. Werner All teleportation and dense coding schemes, J. Phys. A, Volume 34 (2001) no. 35, pp. 7081-7094 (Quantum information and computation) | DOI | MR | Zbl

[44] Arne Winterhof On the non-existence of generalized Hadamard matrices, J. Statist. Plann. Inference, Volume 84 (2000) no. 1-2, pp. 337-342 | DOI | MR | Zbl

Cité par Sources :