Multilevel and multidimensional Hadamard matrices

Authors:
Sarah Spence Adams;Matthew Crawford;Caitlin Greeley;Bryce Lee;Mathav Kishore Murugan
Affiliations:
Franklin W. Olin College of Engineering, Needham, USA;Franklin W. Olin College of Engineering, Needham, USA;Franklin W. Olin College of Engineering, Needham, USA;Franklin W. Olin College of Engineering, Needham, USA;Indian Institute of Technology, Kharagpur, India
Venue:
Designs, Codes and Cryptography
Year:
2009

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Abstract

Multilevel Hadamard matrices (MHMs), whose entries are integers as opposed to the traditional restriction to {卤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.