Cyclic Self-DualZ4-Codes,

Fornodd, theZ4cyclic code generated by 2 is self-dual. We call this a trivial cyclic self-dual code. When do there exist nontrivial cyclic self-dual codes of odd lengthn? We give an answer in this paper by characterizing thesenand describing generators of such codes; this yields an existence test for cyclic difference sets. We also give all examples of nontrivial cyclic self-dual codes up to length 39. From these nontrivial cyclic, self-dual codes, construction A yields unimodular lattices of Type I, some of which are extremal; extension and augmentation yields three new extremal Type II codes of length 32, and an extremal self-dual code of Type II of length 40.