  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.
