On the existence of frames of the Niemeier lattices and self-dual codes over Fp

Motivated by the work of Montague in 1994, we demonstrate that every Niemeier lattice can be constructed as an even unimodular neighbor of the odd unimodular lattice obtained from some self-dual code over Fp by Construction A for each prime number p with 5⩽p⩽23. This implies that every Niemeier lattice contains a 2k-frame for k=2,3,…,28.