Constacyclic codes with best-known parameters

In this paper, we construct several infinite families of $q$-ary constacyclic codes over a finite field ${\mathbb{F}}_q$ with length $n$, dimension around $n\mathrm{/}2$, and minimum distance at least $cn\mathrm{/}{\log }_{q}n$ for some positive constant $c$. They contain many constacyclic codes with optimal, or almost-optimal, or best-known parameters. We also consider constacyclic codes of various lengths.