Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Cryptography and Communications
Published in: Cryptography and Communications 2/2010

01-09-2010

Hadamard matrices and their applications: Progress 2007–2010

Author: K. J. Horadam

Published in: Cryptography and Communications | Issue 2/2010

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

We survey research progress in Hadamard matrices, especially cocyclic Hadamard matrices, their generalisations and applications, made over the past three years. Advances in 20 specific problems and several new research directions are outlined. Two new problems are presented.
previous article Guest editorial for the special issue on design theory
next article Circulant weighing matrices

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
Footnotes
1
Also denoted GH(v, N), or GH(v, m) when N is a group of m th roots of unity.
 
2
The “opposite” multiplication in Aut(N) is needed: σ 1 σ 2 = σ 1 ∘ σ 2.
 
3
The formula in [34, Def. 6.2] is \(\partial\phi(g, h) = \phi(g)^{-1} (\phi(h)^{\varepsilon(g)})^{-1} \phi(g h)\); that is, the coboundary \(\partial(\phi^{-1})\) according to (3). Both definitions of \(\partial\phi\) are in use. Each is correct, if applied consistently.
 
4
A different definition of Hadamard graph (due to Ito [39]) is the graph with vertex set \(V_{4n} = \{0, 1\}^{4n}\) and edge set \(E_{4n} = \{(u, v) \in V_{4n}^2 ~|~ d_H(u, v) = 2n\}\), where d H (u, v) is the Hamming distance between u and v.
 
Literature
1.
Adams, S.S., Crawford, M., Greeley, C., Lee, B., Murugan, M.K.: Multilevel and multidimensional Hadamard matrices. Des. Codes Cryptogr. 51, 245–252 (2009)CrossRefMathSciNet
2.
Alvarez, V., Armario, J.A., Frau, M.D., Gudiel, F., Osuna, A.: Rooted trees searching for cocyclic Hadamard matrices over D 4t , AAECC-19. In: Bras-Amoros, M., Hoholdt, T. (eds.) LNCS, vol. 5527, pp. 204–214. Springer, Berlin (2009)
3.
Alvarez, V., Armario, J.A., Frau, M.D., Real, P.: A system of equations for describing cocyclic Hadamard matrices. c 16(4), 276–290 (2008)MATHMathSciNet
4.
5.
Alvarez, V., Frau, M.D., Osuna, A.: A Heuristic Procedure with Guided Reproduction for Constructing Cocyclic Hadamard Matrices, ICANNGA 2009. In: Kolehmainen, M. et al., (eds.) LNCS, vol. 5495, pp. 150–160. Springer, Berlin (2009)
6.
Bierbrauer, J.: New commutative semifields and their nuclei. In: Bras-Amorós, M., Hoholdt, T. (eds.). AAECC 2009. LNCS, vol. 5527, pp. 179–185 (2009)
7.
8.
Borges, J., Phelps, K.T., Rifa, J.: The rank and kernel of extended 1-perfect \({\mathbb Z}_4\)-linear codes and additive non-\({\mathbb Z}_4\)-linear codes. IEEE Trans. Inf. Theory 49(8), 2028–2034 (2003)CrossRefMathSciNet
9.
Brouwer, A.E., Cohen, A.M., Neumaier, A.: Distance-regular Graphs. Springer, Heidelberg, Berlin (1989)MATH
10.
Budaghyan, L., Carlet, C., Pott, A.: New classes of almost bent and almost perfect nonlinear polynomials. IEEE Trans. Inf. Theory 52, 1141–1152 (2006)MATHCrossRefMathSciNet
11.
Budaghyan, L., Carlet, C., Leander, G.: Another class of quadratic APN binomials over \({\mathbb F}_{2^n}\): the case n divisible by 4. In: Proceedings, International Workshop on Coding and Cryptography, INRIA-Rocquencourt, France, 16–20 April 2007, pp. 49–58
12.
Cameron, P.: Personal Communication. 2 July 2009
13.
Carlet, C., Charpin, P., Zinoviev, V.: Codes, bent functions and permutations suitable for DES-like cryptosystems. Des. Codes Cryptogr. 15, 125–156 (1998)MATHCrossRefMathSciNet
14.
Chen, Y.-Q.: Twisted product of cocycles and factorization of semi-regular relative difference sets. J. Comb. Des. 16, 431–441 (2008)MATHCrossRef
15.
16.
Craigen, R.: Mathematical Reviews. MR2265694 (2008d:05036)
17.
Craigen, R., de Launey, W.: Generalized Hadamard matrices whose transposes are not generalized Hadamard matrices. J. Comb. Des. 17, 456–458 (2009)MATHCrossRef
18.
Craigen, R., de Launey, W.: Personal Correspondence. 27 July 2009
19.
Craigen, R., Holzmann, W.H., Kharaghani, H.: On the asymptotic existence of complex Hadamard matrices. J. Comb. Des. 5, 319–327 (1997)MATHCrossRefMathSciNet
20.
Craigen, R., Kharaghani, H.: Hadamard matrices and Hadamard designs. In: Colbourn, C.J., Dinitz, J.H. (eds.) Handbook of Combinatorial Designs, 2nd edn, pp. 273–280. CRC Press, Boca Raton (2007)
21.
de Caen, D., Mathon, R., Moorhouse, G.E.: A family of antipodal distance-regular graphs related to the classical Preparata codes. J. Algebr. Comb. 4, 317–327 (1995)MATHCrossRef
22.
de Launey, W.: On the asymptotic existence of Hadamard matrices. J. Comb. Theory, A 116, 1002–1008 (2009)MATHCrossRef
23.
24.
de Launey, W., Kharaghani, H.: On the asymptotic existence of cocyclic Hadamard matrices. J. Comb. Theory, A 116, 1140–1153 (2009)MATHCrossRef
25.
de Launey, W., Smith, M.J.: Cocyclic orthogonal designs and the asymptotic existence of cocyclic Hadamard matrices and maximal size relative difference sets with forbidden subgroup of size 2. J. Comb. Theory, A 93, 37–92 (2001)MATHCrossRef
26.
de Launey, W., Stafford, R.M.: Automorphisms of higher-dimensional Hadamard matrices. J. Comb. Des. 16, 507–544 (2008)MATHCrossRef
27.
Dillon, J.F.: Personal Correspondence. 27 February and 15 March 2010
28.
29.
Farmer, D.G., Horadam, K.J.: Presemifield bundles over GF(p 3). In: Proc. ISIT 2008, pp. 2613–2616. IEEE, Toronto (2008)
30.
Farmer, D.G., Horadam, K.J.: A polynomial approach to cocycles over elementary abelian groups. J. Aust. Math. Soc. 85, 177–190 (2008)MATHCrossRefMathSciNet
31.
32.
Gibbons, P.B., Mathon, R.: Enumeration of generalised Hadamard matrices of order 16 and related designs. J. Comb. Des. 17, 119–135 (2009)MATHCrossRefMathSciNet
33.
Horadam, K.J.: A theory of highly nonlinear functions, AAECC-16. In: Fossorier, M. et al., (eds.) LNCS, vol. 3857, pp. 87–100. Springer, Berlin (2006)
34.
Horadam, K.J.: Hadamard Matrices and Their Applications. Princeton University Press, Princeton, NJ (2007)
35.
Horadam, K.J.: EA and CCZ equivalence of functions over GF(2 n ). In: von zur Gathen, J., et al. (eds.) Proc. WAIFI 2008. LNCS, vol. 5130, pp. 134–143. Springer, Berlin (2008)
36.
Horadam, K.J.: Orbiting round the Five-fold constellation (invited lecture). In: Group Theory, Combinatorics and Computation 2009. International Conference in Honour of Professor Cheryl Praeger’s 60th Birthday, Perth, 5 January 2009
37.
Horadam, K.J., Farmer, D.G.: Bundles, presemifields and nonlinear functions. In: Augot, D., Sendrier, N., Tillich J.-P. (eds.) Proc. International Workshop on Coding and Cryptography, pp. 197–206. Versailles, France (2007)
38.
39.
40.
Kyureghyan, G.M., Pott, A.: Some theorems on planar mappings. In: von zur Gathen, J., et al. (eds.) Proc. WAIFI 2008. LNCS, vol. 5130, pp. 117–122. Springer, Berlin (2008)
41.
Ma, K.: Equivalence classes of n-dimensional proper Hadamard matrices. Australas. J. Combin. 25, 3–17 (2002)MATHMathSciNet
42.
McGuire, G., Ward, H.N.: Cocyclic Hadamard matrices from forms over finite Frobenius rings. Linear Algebra App. 430, 1730–1738 (2009)MATHCrossRefMathSciNet
43.
Ó Catháin P.: Group Actions on Hadamard Matrices, M. Litt. Thesis, National University of Ireland, Galway (2008)
44.
45.
46.
Phelps, K.T., Rifa, J., Villanueva, M.: Rank and kernel of binary Hadamard codes. IEEE Trans. Inf. Theory 51(11), 3931–3937 (2005)CrossRefMathSciNet
47.
Phelps, K.T., Rifa, J., Villanueva, M.: Hadamard codes of length 2 t s (s odd): Rank and kernel, AAECC-16. In: Fossorier, M., et al. (eds.) LNCS, vol. 3857, pp. 328–337. Springer, Berlin (2006)
48.
Pott, A.: A survey on relative difference sets. In: Groups, Difference Sets and the Monster, pp. 195–232. de Gruyter, New York (1996)
49.
Rifa, J., Solov’eva, F.I., Villanueva, M.: On the intersection of \({\mathbb Z}_2{\mathbb Z}_4\)-additive Hadamard codes. IEEE Trans. Inf. Theory 55(4), 1766–1774 (2009)CrossRefMathSciNet
50.
51.
Trinh, Q.K., Fan, P., Gabidulin, E.M.: Multilevel Hadamard matrices and zero correlation zone sequences. Electron. Lett. 42(13), 748–750 (2006)CrossRef
52.
Yang, Y.X., Niu, X.X., Xu, C.Q.: Theory and Applications of Higher Dimension Hadamard Matrices, 2nd edn. Chapman and Hall (2010). ISBN: 978-1-4398180-7-7
53.
Metadata
Title
Hadamard matrices and their applications: Progress 2007–2010
Author
K. J. Horadam
Publication date
01-09-2010
Publisher
Springer US
Published in
Cryptography and Communications / Issue 2/2010
Print ISSN: 1936-2447
Electronic ISSN: 1936-2455
DOI
https://doi.org/10.1007/s12095-010-0032-0

Other articles of this Issue 2/2010

Cryptography and Communications 2/2010 Go to the issue

OriginalPaper

Exotic complex Hadamard matrices and their equivalence

OriginalPaper

The problem of mutually unbiased bases in dimension 6

OriginalPaper

Circulant weighing matrices

OriginalPaper

A Fourier-analytic approach to counting partial Hadamard matrices

OriginalPaper

Combinatorial bounds and characterizations of splitting authentication codes

OriginalPaper

Mutually orthogonal Latin squares and mutually unbiased bases in dimensions of odd prime power

Premium Partner

    Image Credits