Improved Multilevel Hadamard Matrices and their Generalizations over the Gaussian and Hamiltonian Integers

Abstract

Multilevel Hadamard matrices (MHMs), whose entries are integers as opposed to the traditional restriction to $\{\pm 1\}$, have been introduced as a way to construct multilevel zero-correlation zone sequences for use in approximately synchronized code division multiple access (AS-CDMA) systems. This paper provides a construction technique to produce $2^m\times 2^m$ MHMs whose $2^m$ alphabet entries form an arithmetic progression, up to sign. This construction improves upon existing constructions because it permits control over the spacing and overall span of the MHM entries. MHMs with such regular alphabets are a more direct generalization of traditional Hadamard matrices and are thus expected to be more useful in applications analogous to those of Hadamard matrices. This paper also introduces mixed-circulant MHMs which provide a certain advantage over known circulant MHMs of the same size.

MHMs over the Gaussian (complex) and Hamiltonian (quaternion) integers are introduced. Several constructions are provided, including a generalization of the arithmetic progression construction for MHMs over real integers. Other constructions utilize amicable pairs of MHMs and c-MHMs, which are introduced as natural generalizations of amicable orthogonal designs and c-Hadamard matrices, respectively. The constructions are evaluated against proposed criteria for interesting and useful MHMs over these generalized alphabets.