We have seen that one of the basic Hadamard matrix building methods is based on the construction of a class of "special-component" matrices that can be plugged into arrays (templates) to generate Hadamard matrices. In Chapter 4, we discussed how to construct these special-component matrices. In this chapter, we focus on the second component of the plug-in template method: construction of arrays/templates. Generally, the arrays into which suitable matrices are plugged are orthogonal designs (ODs), which have formally orthogonal rows (and columns).

The theory of ODs dates back over a century. ODs have several variations such as the Goethals-Seidel arrays and Wallis-Whiteman arrays. Numerous approaches for construction of these arrays/templates have been developed. A survey of OD applications, particularly space-time block coding, can be found in Refs. 3-7, 23, 24, 34, 87-91.

The space-time block codes are particularly attractive because they can provide full transmit diversity while requiring a very simple decoupled maximumlikelihood decoding method. The combination of space and time diversity has moved the capacity of wireless communication systems toward theoretical limits; this technique has been adopted in the 3G standard in the form of an Alamouti code and in the newly proposed standard for wireless LANs IEEE 802.11n.

In this chapter, two plug-in template methods of construction of Hadamard matrices are presented. We focus on construction of only Baumert-Hall, Plotkin, and Welch arrays, which are the subsets of ODs.

9.1 ODs

The original definition of OD was proposed by Geramita et al. Dr. Seberry (see Fig. 9.1), a co-author of that paper, is world renowned for her discoveries on Hadamard matrices, ODs, statistical designs, and quaternion OD (QOD). She also did important work on cryptography. Her studies of the application of discrete mathematics and combinatorial computing via bent functions and S-box design have led to the design of secure crypto algorithms and strong hashing algorithms for secure and reliable information transfer in networks and telecommunications. Her studies of Hadamard matrices and ODs are also applied in CDMA technologies.