The author generalizes the definition of higher power residue codes introduced in [\textit{R. Chapman}, Finite Fields Appl. 3, 353--369 (1997; Zbl 1053.94559)] for codes over fields to codes over \(\mathbb{Z}_4\). Using codes over the ring \(\mathbb{Z}_4[\omega]\), where \(\omega^2+\omega +1=0\), a character of order 6, and quadratic residue codes, the author constructs self-dual codes of length \(3(p+1)\) over \(\mathbb{Z}_4\). The most interesting example is a code which produces the Leech lattice. This code has an identical symmetrized weight enumerator to a code given in [\textit{A. Bonnecaze, P. Solé} and \textit{A. R. Calderbank}, IEEE Trans. Inf. Theory 41, 366--377 (1995; Zbl 0822.94009)] which also produces the Leech lattice, but the codes are shown to be inequivalent.