Cyclic self-dual Z4-codes

Binary cyclic self-dual codes of odd length do not exist. It is not the case, however, of Z₄-cyclic self-dual codes. Some of the first examples, aside from the trivial self-dual code, are supplemented quadratic residue codes. The aim of this paper is to characterize arithmetically the (odd) lengths where such non-trivial cyclic self-dual Z₄-codes can exist and to give some examples for short lengths. As the length is odd, it is hopeless to try to obtain directly Type II codes; i.e. codes whose Euclidean weights are multiples of 8 since these exist only for lengths a multiples of 8. It is possible, nonetheless, to obtain Type I Z₄-codes, and, accordingly, by construction A, Type I lattices. We obtain, in that way, the only two extremal odd lattices in dimensions 15-47: the shorter Leech lattice O₂₃ in dimension 23 and A₁₅⁺ , in dimension 15. Invariant theory enables us to compute the symmetric weight enumerators of the codes and therefore the theta series of the associated lattices, including the norm and kissing number thereof. In passing, we mention an amusing non-existence arithmetic criterion for cyclic projective planes