Designs, Codes and Cryptography

, Volume 51, Issue 3, pp 245–252 | Cite as

Multilevel and multidimensional Hadamard matrices

  • Sarah Spence Adams
  • Matthew Crawford
  • Caitlin Greeley
  • Bryce Lee
  • Mathav Kishore Murugan


Multilevel Hadamard matrices (MHMs), whose entries are integers as opposed to the traditional restriction to {±1}, were introduced by Trinh, Fan, and Gabidulin in 2006 as a way to construct multilevel zero-correlation zone sequences, which have been studied for use in approximately synchronized code division multiple access systems. We answer the open question concerning the maximum number of distinct elements permissible in an order n MHM by proving the existence of an order n MHM with n elements of distinct absolute value for all n. We also define multidimensional MHMs and prove an analogous existence result.


Hadamard matrix Multilevel Hadamard matrix Multidimensional matrix Circulant matrix Zero-correlation zone sequence 

Mathematics Subject Classifications (2000)

05B20 94A99 


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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Sarah Spence Adams
    • 1
  • Matthew Crawford
    • 1
  • Caitlin Greeley
    • 1
  • Bryce Lee
    • 1
  • Mathav Kishore Murugan
    • 2
  1. 1.Franklin W. Olin College of EngineeringNeedhamUSA
  2. 2.Indian Institute of TechnologyKharagpurIndia

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