Construction of self-dual codes over $\mathbb {Z}_{2^{m}}$

Abstract

Self-dual codes (Type I and Type II codes) play an important role in the construction of even unimodular lattices, and hence in the determination of Jacobi forms. In this paper, we construct Type I and Type II codes (of higher lengths) over the ring $\mathbb {Z}_{2^{m}}$ of integers modulo 2^m from shadows of Type I codes over $\mathbb {Z}_{2^{m}}$, and obtain their complete weight enumerators. As an application, we determine some Jacobi forms on the modular group ${\Gamma }(1) = SL(2,\mathbb {Z})$. Besides this, we construct self-dual codes (of higher lengths) over $\mathbb {Z}_{2^{m}}$ from the generalized shadow of a self-dual code $\mathcal {C}$ of length n over $\mathbb {Z}_{2^{m}}$ with respect to a vector $s \in \mathbb {Z}_{2^{m}}^{n} \setminus \mathcal {C}$ satisfying either s ⋅ s ≡ 0 (mod 2^m) or s ⋅ s ≡ 2^m−1 (mod 2^m). We also illustrate our results with some examples.