In this chapter, we introduce constructions for signal sets with low crosscorrelation. These sequences have important applications in wireless CDMA communications. There are three classic constructions for signal sets with low correlation, namely, the Gold-pair construction, the Kasami (small) set construction, and the bent function signal set construction. In Section 10.1, we introduce some basic concepts and properties for crosscorrelation of sequences or functions, signal sets, and one-to-one correspondences among sequences, polynomial functions, and boolean functions. After that, three classic constructions will be presented in Sections 9.2, 9.3, and 9.4 respectively. With the development of new technologies, the demand for constraints on other parameters, such as linear spans of sequences, and the sizes of the signal sets has increased. Here, we will provide two examples of constructions that sacrifice ideal correlation in order to improve other properties, in Sections 9.5 and 9.6, respectively. One example is the interleaved construction for large linear spans, and the other is ℤ4 sequences to obtain large sizes of signal sets.

Crosscorrelation, signal sets, and boolean functions

In this section, we discuss some basic properties of crosscorrelation of sequences (some of them have been discussed in Chapter 1), refine the concept of signal sets, and develop the one-to-one correspondence between sequences and boolean functions. (Note that the one-to-one correspondence between sequences and functions is discussed in Chapter 6.)

We will keep the following notation in this section.