  An anticode ${\bf C} \subset {\bf F}_q^n$ with the diameter $\delta$ is a
code in ${\bf F}_q^n$ such that the distance between any two distinct codewords
in ${\bf C}$ is at most $\delta$. The famous Erd\"{o}s-Kleitman bound for a
binary anticode ${\bf C}$ of the length $n$ and the diameter $\delta$ asserts
that $$|{\bf C}| \leq \Sigma_{i=0}^{\frac{\delta}{2}} \displaystyle{n \choose
i}.$$ In this paper, we give an antiGriesmer bound for $q$-ary projective
linear anticodes, which is stronger than the above Erd\"{o}s-Kleitman bound for
binary anticodes. The antiGriesmer bound is a lower bound on diameters of
projective linear anticodes. From some known projective linear anticodes, we
construct some linear codes with optimal or near optimal minimum distances. A
complementary theorem constructing infinitely many new projective linear
$(t+1)$-weight code from a known $t$-weight linear code is presented. Then many
new optimal or almost optimal few-weight linear codes are given and their
weight distributions are determined. As a by-product, we also construct several
infinite families of three-weight binary linear codes, which lead to
$l$-strongly regular graphs for each odd integer $l \geq 3$.
