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 $l \geq 3$.

翻译：反码 ${\bf C} \subset {\bf F}_q^n$ 是一个定义在 ${\bf F}_q^n$ 上的码，其直径为 $\delta$，即该码中任意两个不同码字之间的距离至多为 $\delta$。对于长度为 $n$、直径为 $\delta$ 的二元反码 ${\bf C}$，著名的 Erdős-Kleitman 界断言：$$|{\bf C}| \leq \Sigma_{i=0}^{\frac{\delta}{2}} \displaystyle{n \choose i}.$$ 本文针对 $q$ 元射影线性反码给出了一个反 Griesmer 界，该界强于上述针对二元反码的 Erdős-Kleitman 界。反 Griesmer 界是射影线性反码直径的一个下界。基于一些已知的射影线性反码，我们构造了一些具有最优或接近最优最小距离的线性码。本文提出了一个互补定理，可以从一个已知的 $t$ 重线性码构造出无穷多个新的射影线性 $(t+1)$ 重码。随后，我们给出了许多新的最优或几乎最优的少重线性码，并确定了它们的重量分布。作为副产品，我们还构造了多个无限族的三重二元线性码，这些码为每个大于等于 3 的奇数 $l$ 导出了 $l$-强正则图。