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 A₂₄ and D₂₄. The lattices are characterised by means of a sublattice isomorphic to the root lattice of type Aₙ₋₁. A sufficient condition for existence of an orthogonal k-frame of such a lattice is given in terms of symmetric $2$-designs.
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Matsuo et al. (2024) studied this question.
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