Extremal Type I ${\mathbb{Z}}_k$-codes and $k$-frames of odd unimodular lattices

For some extremal (optimal) odd unimodular lattice $L$ in dimensions $12,16,20,28,32,36,40$ and $44$, we determine all integers $k$ such that $L$ contains a $k$-frame. This result yields the existence of an extremal Type I ${\mathbb{Z}}_k$-code of lengths $12,16,20,32,36,40$ and $44$, and a near-extremal Type I ${\mathbb{Z}}_k$-code of length $28$ for positive integers $k$ with only a few exceptions.