On the Existence of Frames of Some Extremal Odd Unimodular Lattices and Self-Dual Zk-Codes

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