Quaternion orthogonal designs from complex companion designs

The success of applying generalized complex orthogonal designs as space–time block codes recently motivated the definition of quaternion orthogonal designs as potential building blocks for space–timepolarization block codes. This paper offers techniques for constructing quaternion orthogonal designs via combinations of specially chosen complex orthogonal designs. One technique is used to build quaternion orthogonal designs on complex variables for any even number of columns. A second related technique is applied to maximum rate complex orthogonal designs to generate an infinite family of quaternion orthogonal designs on complex variables such that the resulting designs have no zero entries. This second technique is also used to generate an infinite family of quaternion orthogonal designs defined over quaternion variables that display a regular redundancy. The proposed constructions are theoretically important because they provide the first known direct techniques for building infinite families of orthogonal designs over quaternion variables for any number of columns. © 2007 Elsevier Inc. All rights reserved. ∗ Corresponding author. Tel.: +1 781 292 2536; fax: +1 781 292 2505. E-mail addresses: [email protected] (S.S. Adams), [email protected] (J. Seberry), nathaniel.karst@alumni. olin.edu (N. Karst), [email protected] (J. Pollack), [email protected] (T.A. Wysocki). 1 Supported in part by NSA Grant H98230-07-1-0022 and an NSF-AWM Mentoring Travel Grant. 2 Current address: Peter Kiewit Institute, University of Nebraska-Lincoln, Omaha, NE 68588, USA. 0024-3795/$ see front matter ( 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2007.09.013 S.S. Adams et al. / Linear Algebra and its Applications 428 (2008) 1056–1071 1057 AMS classification: Primary 05B20; Secondary 94A05, 62K05, 62K10