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Construction of self-dual codes over \(\mathbb {Z}_{2^{m}}\). (English) Zbl 1344.94099

Summary: 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 \({\Gamma }(1) = \mathrm{SL}(2,\mathbb {Z})\). 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 \(s\cdot s \equiv 0 \pmod {2^{m}}\) or \(s\cdot s \equiv 2^{m-1} \pmod {2^{m}}\). We also illustrate our results with some examples.

MSC:

94B15 Cyclic codes
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References:

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