  An $m$-$cover$ of lines of a finite projective space ${\rm PG}(r,q)$ (of a
finite polar space $\cal P$) is a set of lines $\cal L$ of ${\rm PG}(r,q)$ (of
$\cal P$) such that every point of ${\rm PG}(r,q)$ (of $\cal P$) contains $m$
lines of $\cal L$, for some $m$. Embed ${\rm PG}(r,q)$ in ${\rm PG}(r,q^2)$.
Let $\bar{\cal L}$ denote the set of points of ${\rm PG}(r,q^2)$ lying on the
extended lines of $\cal L$.
  An $m$-cover $\cal L$ of ${\rm PG}(r,q)$ is an $(r-2)$-dual $m$-cover if
there are two possibilities for the number of lines of $\cal L$ contained in an
$(r-2)$-space of ${\rm PG}(r,q)$. Basing on this notion, we characterize
$m$-covers $\cal L$ of ${\rm PG}(r,q)$ such that $\bar{\cal L}$ is a
two-character set of ${\rm PG}(r,q^2)$. In particular, we show that if $\cal L$
is invariant under a Singer cyclic group of ${\rm PG}(r,q)$ then it is an
$(r-2)$-dual $m$-cover.
  Assuming that the lines of $\cal L$ are lines of a symplectic polar space
${\cal W}(r,q)$ (of an orthogonal polar space ${\cal Q}(r,q)$ of parabolic
type), similarly to the projective case we introduce the notion of an
$(r-2)$-dual $m$-cover of symplectic type (of parabolic type). We prove that an
$m$-cover $\cal L$ of ${\cal W}(r,q)$ (of ${\cal Q}(r,q)$) has this dual
property if and only if $\bar{\cal L}$ is a tight set of an Hermitian variety
${\cal H}(r,q^2)$ or of ${\cal W}(r,q^2)$ (of ${\cal H}(r,q^2)$ or of ${\cal
Q}(r,q^2)$). We also provide some interesting examples of $(4n-3)$-dual
$m$-covers of symplectic type of ${\cal W}(4n-1,q)$.
