  A series of integral lattices parametrised by integers $k,m,n$ are introduced
and investigated, where $n$ is the rank of the lattice, including the root
lattices described in a uniform way and unimodular lattices such as the
Niemeier lattices of type $A_{24}$ and $D_{24}$. The lattices are characterised
by means of a sublattice isomorphic to the root lattice of type $A_{n-1}$. A
sufficient condition for existence of an orthogonal $k$-frame of such a lattice
is given in terms of symmetric $2$-designs.
