Some Properties of Binary Matrices and Quasi-Orthogonal Signals Based on Hadamard Equivalence

Ki-Hyeon PARK  Hong-Yeop SONG  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E95-A   No.11   pp.1862-1872
Publication Date: 2012/11/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E95.A.1862
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Signal Design and Its Applications in Communications)
Category: Sequences
Hadamard equivalence,  orthogonality,  Quasi-orthogonal signal,  Quasi-Hadamard matrix,  

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We apply the Hadamard equivalence to all the binary matrices of the size mn and study various properties of this equivalence relation and its classes. We propose to use HR-minimal as a representative of each equivalence class, and count and/or estimate the number of HR-minimals of size mn. Some properties and constructions of HR-minimals are investigated. Especially, we figure that the weight on an HR-minimal's second row plays an important role, and introduce the concept of Quasi-Hadamard matrices (QH matrices). We show that the row vectors of mn QH matrices form a set of m binary vectors of length n whose maximum pairwise absolute correlation is minimized over all such sets. Some properties, existence, and constructions of Quasi-orthogonal sequences are also discussed. We also give a relation of these with cyclic difference sets. We report lots of exhaustive search results and open problems, one of which is equivalent to the Hadamard conjecture.