A systematic search for optimal lattice rules of specified trigonometric degree
d
d
over the hypercube
[
0
,
1
)
s
[0,1)^s
has been undertaken. The search is restricted to a population
K
(
s
,
δ
)
K(s,\delta )
of lattice rules
Q
(
Λ
)
Q(\Lambda )
. This includes those where the dual lattice
Λ
⊥
\Lambda ^\perp
may be generated by
s
s
points
h
\bf h
for each of which
|
h
|
=
δ
=
d
+
1
|\textbf {h} | = \delta =d+1
. The underlying theory, which suggests that such a restriction might be helpful, is presented. The general character of the search is described, and, for
s
=
3
s=3
,
d
≤
29
d \leq 29
and
s
=
4
s=4
,
d
≤
23
d \leq 23
, a list of
K
K
-optimal rules is given. It is not known whether these are also optimal rules in the general sense; this matter is discussed.