  Consider a class of simplices defined by systems $A x \leq b$ of linear
inequalities with $\Delta$-modular matrices. A matrix is called
$\Delta$-modular, if all its rank-order sub-determinants are bounded by
$\Delta$ in an absolute value. In our work we call a simplex $\Delta$-modular,
if it can be defined by a system $A x \leq b$ with a $\Delta$-modular matrix
$A$. And we call a simplex empty, if it contains no points with integer
coordinates. In literature, a simplex is called lattice-simplex, if all its
vertices have integer coordinates. And a lattice-simplex called empty, if it
contains no points with integer coordinates excluding its vertices.
  Recently, assuming that $\Delta$ is fixed, it was shown that the number of
$\Delta$-modular empty simplices modulo the unimodular equivalence relation is
bounded by a polynomial on dimension. We show that the analogous fact holds for
the class of $\Delta$-modular empty lattice-simplices. As the main result,
assuming again that the value of the parameter $\Delta$ is fixed, we show that
all unimodular equivalence classes of simplices of the both types can be
enumerated by a polynomial-time algorithm. As the secondary result, we show the
existence of a polynomial-time algorithm for the problem to check the
unimodular equivalence relation for a given pair of $\Delta$-modular, not
necessarily empty, simplices.
