In Figure 6a--f
we present a few examples to illustrate
this behavior. These
examples depict the result of generating all pairs of consecutive
numbers in the
period of full-period LCGs. For each modulus, the plotted points show the
effect of our choice of the multiplier, **a**. In Figure 6a
we show the set of
points produced by scaling by 1/509 the output of the
generator .
Note that the points form more than one set of rows.
That is, there are several
angles from which the points appear to line up. From some perspectives, the
rows are close together, while from at least one perspective, the rows are
rather far apart. The best situation is to have the maximum spacing of the
rows, when viewed over all angles, as small as possible. When the maximum
spacing is large, as it certainly is in Figure 6b,
, then
clearly the unit square is not well covered by this set of points, and the
results of our simulation may be adversely affected by
this ``striping.'' Figure 6c
shows the effect of a good choice of **a**, , for which the
maximum spacing between rows is clearly smaller than it is in the first two
cases, and where the points cover the unit square with nearly optimal
uniformity, given that with this generator we can generate only a small number
of points.

Figures 6a-c
are examples that all use the prime modulus **m = 509**. When **m**
is a power of two, the other major case to consider, the situation is similar,
as depicted in Figures 6d and e.
Recall that when **m** is a power of two
and **c = 0**, the full period is , but only if mod 8 or mod 8. Thus we choose
so that with period 512, we
plot almost the same number of points (508) as we see in Figures
6a-c. Figure 6d
illustrates the result of using a good choice of **a** (
mod 8). As in the prime modulus case, the plotted points form a
uniform looking
lattice. But when we plot the points in Figure 6e
generated by taking mod 8, we see that the
pattern looks like two lattice structures
slightly offset from one another. This double lattice distinguishes the mod 8 case from the mod 8 case and is the reason that mod 8 is preferred, even though both sets of **a** values produce the
same period. As an aside, the two lattice structures in the mod 8
case come about from plotting separately the pairs beginning with odd indices
and those beginning with even indices.