## JPL's Wireless Communication Reference Website## Chapter: Analog and Digital Transmission |

If the LSFRs are chosen appropriately, Gold sequences have better cross-correlation properties than maximum length LSFR sequences.

if *l*
is odd,
*t* = 2^{(l+1)/2} + 1, and

if *l* is even,
*t* = 2^{(l+2)/2} + 1.

Thus, a Gold sequence formally is an arbitrary phase of a sequence in the set* G(u,v)* defined by

*G(u,v)*= {*u,v,u * v, u * Tv, u *T*^{2}* v, U * T*^{(N-1)} *v*}

*T*^{k} denotes the operator which shifts vectors cyclically to the left by* k* places, *** is the exclusive OR operator and *u*,* v* are *m*-sequences of period generated by different primitive binary polynomials.

It is well known that the "partial crosscorrelation" values can be altered by changing the phases of
the code sequences. In theory, then, it is possible to find optimal phases which minimize the interference in the desired data signal.
However,
for* K* users each employing a sequence of period* N*,
there are a total of *N K* different sets of sequence phases
possible. For a realistic system, e.g. , direct computation becomes intractable.
Even when direct computation is performed, the reduction
in interference of the optimal set of phases
over the worst set of phases is 30%.