  We seek shifted lattice rules that are good for high dimensional integration
over the unit cube in the setting of an unanchored weighted Sobolev space of
functions with square-integrable mixed first derivatives. Many existing studies
rely on random shifting of the lattice, whereas here we work with lattice rules
with a deterministic shift. Specifically, we consider "half-shifted" rules, in
which each component of the shift is an odd multiple of $1/(2N)$, where $N$ is
the number of points in the lattice. We show, by applying the principle that
\emph{there is always at least one choice as good as the average}, that for a
given generating vector there exists a half-shifted rule whose squared
worst-case error differs from the shift-averaged squared worst-case error by a
term of order only ${1/N^2}$. Numerical experiments, in which the generating
vector is chosen component-by-component (CBC) as for randomly shifted lattices
and then the shift by a new "CBC for shift" algorithm, yield encouraging
results.
