In this chapter we briefly review some of the sequences that are related to m-sequences, many of which have found applications in communications. See also Section 15.5, where the linear span of sequences derived from m-sequences is discussed. Further information on these and many other sequences may be found in Helleseth and Kumar's survey article [88], Cusick, Ding, and Renvall's monograph [35], and Golomb and Gong's textbook [63].

Welch bound

For applications to spread spectrum communications, one attempts to find a collection of shift distinct sequences with low pairwise cross-correlation values. For a given (maximal) cross-correlation, there are theoretical limitations on the number of sequences in such a collection, the simplest of which is the Welch bound [204].

Suppose we have a collection of n periodic sequences of elements in a finite field F, each with the same period T. We can expand this set to include all the shifts of these sequences. If T is the minimal period of each sequence and the sequences are pairwise shift distinct, then we obtain a set of N = Tn vectors, commonly referred to as a signal set. Let us denote these vectors by a(1), a(2), …, a(N). Let χ : F → ℂ× be a nontrivial character.