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 Z
_k
-code of lengths 12, 16, 20, 32, 36, 40, and 44, and a near-extremal Type I Z
_k
-code of length 28 for positive integers k with only a few exceptions.