Type II self-dual codes over finite rings and even unimodular lattices (Q1301859)

Self-dual codes over rings \({\mathbb Z}_k\) are studied, with special attention given to the case \(k=2^m\). A Type II code over \({\mathbb Z}_{2^m}\) is a self-dual code in which every codeword has Euclidean weight a multiple of \(2^{m+1}\). Such codes are shown to be related to self-orthogonal codes over \({\mathbb Z}_{2^{m+1}}\), and to even unimodular lattices, generalising a result of \textit{A. Bonnecazé, P. Solé} and \textit{A. R. Calderbank} [IEEE Trans. Inf. Theory 41, 366-377 (1995; Zbl 0822.94009)]. The existence of double circulant Type II codes over \({\mathbb Z}_{2^m}\) is considered and examples are given of extremal codes for \(m=3,4,5\).