  The class of $\ell$-maximum distance separable ($\ell$-MDS) codes {is a}
generalization of maximum distance separable (MDS) codes {that} has attracted a
lot of attention due to its applications in several areas such as secret
sharing schemes, index coding problems, informed source coding problems, and
combinatorial $t$-designs. In this paper, for $\ell=1$, we completely solve a
conjecture recently proposed by Heng $et~al.$ (Discrete Mathematics, 346(10):
113538, 2023) and obtain infinite families of $1$-MDS codes with general
dimensions holding $2$-designs. These later codes are also been proven to be
optimal locally recoverable codes. For general {positive integers} $\ell$ and
$\ell'$, we construct new $\ell$-MDS codes from known $\ell'$-MDS codes via
some classical propagation rules involving the extended, expurgated, and
$(u,u+v)$ constructions. Finally, we study some general results including
characterization, weight distributions, and bounds on maximum lengths of
$\ell$-MDS codes, which generalize, simplify, or improve some known results in
the literature.
