11

the same number of storage elements. However, if a large number of sequences is required for a

given application, and if the cross correlation function is more important than the autocorrelation

function, then the Gold and Kasami sequences are much better than m-sequences [13].

# Gold sequences take a pair of m-sequences with sequences of length n are generated by taking

the modulo-2 sum of one (called a) with the n cyclical shifted version of the other (called b).

# This generates 2^{n }+ 1 different sequences each period 2^{n }- 1 and such that the cross-correlation

function ϕ(a,b) of any pair of such sequences satisfies the equation [14]:

2

+1 /2

+ 1,

2

+2 /2

+ 1,

(3)

(, ) =

# Welsh showed that the maximum cross-correlation between any two sequences in a set length N

sequences of cardinality M is lower bounded [15]. Specifically, he showed that the maximum

cross-correlation between two sequences is lower bounded by

M − 1 / MN − 1

, where M is

the number of codes in the set. For relatively large sets it can be concluded that the maximum

cross-correlation is greater than

1/N.

By applying these bounds to Gold sequences (M = 2^{n }+

1, N = 2^{n }– 1), it can be seen that the max cross-correlation is [16]:

(4)

≈

2/ , 4/ ,