skip to main content
10.1145/2381716.2381720acmotherconferencesArticle/Chapter ViewAbstractPublication PagescubeConference Proceedingsconference-collections
research-article

Revisiting butson hadamard matrix construction based on a prominent class of pseudo-random sequences

Published: 03 September 2012 Publication History

Abstract

We revisit the well known matrix construction method based on one of the most popular classes of p --- ary pseudo-random sequences. Our detailed considerations explicitly show that both the ideal autocorrelation property and the balancedness are required to construct normalized Butson Hadamard matrices through that method. The refinement made is illustrated by some examples.

References

[1]
M. Antweiler and L. Bomer, Complex sequences over GF(pm) with a two-level autocorrelation function and a large linear span, IEEE Trans. on Information Theory, vol. 38, pp. 120--130, Jan. 1992.
[2]
Y.L. Borissov, M. H. Lee, and S. P. Balakannan, Jacket matrix construction using M-sequences, isitc, pp.333--337, 2007 International Symposium on Information Technology Convergence (ISITC 2007), 2007.
[3]
A.T. Butson, Generalized Hadamard matrices, Proc. Am. Math. Soc., vol. 13, "pp. 894--898, 1962.
[4]
M. Cohn and A. Lempel, On fast m-sequence transforms, IEEE Trans. on Information Theory, vol. 23, pp. 135--137, Jan. 1977.
[5]
C.H. Cooke and I. Heng, Polynomial construction of complex Hadamard matrices with cyclic core, Applied Mathematics Letters, vol. 12, pp. 87--93, 1999.
[6]
P. Fan and M. Darnell, Sequence design for communications applications, John Wiley & Sons Inc., New York, 1996.
[7]
G. Garg, T. Helleseth, and P. V. Kumar, Recent advances in low-correlation sequences, Chapter 3.2 in, New directions in wireless communications research, V. Tarokh (ed.), Springer Science+Business Media, LLC 2009.
[8]
S.W. Golomb, Shift register sequences, Holden-Day, San Francisco, CA, 1967; S. W. Golomb, Shift register sequences, revised ed., Aegean Park Press, Laguna Hills, CA, 1982.
[9]
J. Hadamard, Resolution d'une question relative aux determinants, Bulletin des Sciences Mathématiques, 17(2):241--246, 1893.
[10]
T. Helleseth and G. Gong, New nonbinary sequences with ideal two-level autocorrelation, IEEE Trans. on Information Theory, vol. 48, pp. 2868--2872, Nov. 2002.
[11]
T. Helleseth, P. V. Kumar and H. M. Martinsen, A new family of ternary sequences with ideal two-level autocorrelation function, Des., Codes and Cryptogr., vol. 23, pp. 157--166, 2001.
[12]
K.J. Horadam, Hadamard matrices and their applications, New Jersey: Princeton University Press, 2007
[13]
J.W. Jang, J. S. No, and H. Chung, Butson Hadamard matrices with partially cyclic core, Des., Codes and Cryptogr., vol. 43, pp. 93--101, 2007.
[14]
A. Klapper, A. H. Chan, and M. Goresky, Cascaded GMW sequences, IEEE Trans. on Information Theory, vol. 39, pp. 177--183, Jan. 1993.
[15]
M.H. Lee, The center weighted Hadamard transform, IEEE Trans. on CAS-II, vol. 36, no 9, pp. 1247--1249, Sept. 1989.
[16]
S. Matsufuji and P. Fan, Constructions of factorizable multilevel Hadamard matrices, IEICE Trans. Fundamentals, vol. E92-A, no 12, pp.3404--3406, Dec. 2009.
[17]
F.J. McWilliams and N. J. A. Sloane, Pseudo-random sequences and arrays, Proceedings of the IEEE, vol. 64, pp. 1715--1729, Dec. 1976.
[18]
J.S. No, New cyclic difference sets with Singer parameters constructed from d-homogeneous functions, Des., Codes and Cryptogr., vol. 33, pp. 199--213, 2004.
[19]
J.S. No, S. W. Golomb, G. Gong, H. K. Lee, and P. Gaal, Binary pseudorandom sequences of period 2n --- 1 with ideal autocorrelation, IEEE Trans. on Information Theory, vol. 44, pp. 814--817, March 1998.
[20]
J.S. No, New cyclic difference sets with Singer parameters constructed from d-homogeneous functions, Des., Codes and Cryptogr., vol. 33, pp. 199--213, 2004.
[21]
R.A. Scholtz, and L. R. Welch, GMW sequences, IEEE Trans. on Information Theory, vol. 30, pp. 548--553, May 1984.
[22]
J. Seberry and M. Yamada, Hadamard matrices, sequences and block designs, in Contemporary Design Theory - a Collection of Surveys, D. J. Stinson and J. Dinitz, Eds. Wiley, pp. 431--560, 1992.
[23]
J.J. Sylvester, Thoughts on inverse orthogonal matrices, simultaneous sign successions, a tesselated pavements in two or more colours, with applications to Newton's rule, ornamental tile-work, and the theory of numbers, Phil. Mag., vol. 34, pp. 461--475, 1867.
[24]
F. Szollosi, Parameterizating complex Hadamard matrices, Eur. J. Comb., vol. 29, no 5, pp. 1219--1234, July 2007.
[25]
R.J. Turyn, Complex Hadamard matrices, in Combinatorial structures and their applications, pp. 435--437, 1970.
[26]
R.K. Yarlagadda and J. E. Hershey, Hadamard matrix analysis and synthesis with applications to communications and signal/image processing, Norwel, MA: Kluwer, 1997.

Recommendations

Comments

Information & Contributors

Information

Published In

cover image ACM Other conferences
CUBE '12: Proceedings of the CUBE International Information Technology Conference
September 2012
879 pages
ISBN:9781450311854
DOI:10.1145/2381716
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Sponsors

  • CUOT: Curtin University of Technology

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 03 September 2012

Permissions

Request permissions for this article.

Check for updates

Author Tags

  1. Butson-type Hadamard matrix
  2. balancedness
  3. ideal two-level autocorrelation

Qualifiers

  • Research-article

Conference

CUBE '12
Sponsor:
  • CUOT

Contributors

Other Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

  • 0
    Total Citations
  • 74
    Total Downloads
  • Downloads (Last 12 months)0
  • Downloads (Last 6 weeks)0
Reflects downloads up to 03 Mar 2025

Other Metrics

Citations

View Options

Login options

View options

PDF

View or Download as a PDF file.

PDF

eReader

View online with eReader.

eReader

Figures

Tables

Media

Share

Share

Share this Publication link

Share on social media