  In this paper, we define $m$-tail reflexive sheaves as reflexive sheaves on
projective spaces with the simplest possible cohomology. We prove that the rank
of any $m$-tail reflexive sheaf $\mathcal{E}$ on $\mathcal{P}^n$ is greater or
equal to $ nm-m$. We completely describe $m$-tail reflexive sheaves on
$\mathcal{P}^n$ of minimal rank and we construct huge families of $m$-tail
reflexive sheaves of higher rank.
