Let G be a nontrivial abelian group and let A1,…,Ah\documentclass[12pt]{minimal}
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\begin{document}$$A_1,\, \ldots ,\,A_h$$\end{document} (h≥2\documentclass[12pt]{minimal}
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\begin{document}$$h\ge 2$$\end{document}) be nonempty subsets of G. We say that A1,…,Ah\documentclass[12pt]{minimal}
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\begin{document}$$A_1,\, \ldots ,\,A_h$$\end{document} is a complete decomposition of G of order h if A1+⋯+Ah=G\documentclass[12pt]{minimal}
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\begin{document}$$A_1+ \cdots +A_h =G$$\end{document} and Ai∩Aj=∅\documentclass[12pt]{minimal}
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\begin{document}$$A_i\cap A_j=\emptyset $$\end{document} for i,j=1,…,h\documentclass[12pt]{minimal}
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\begin{document}$$i,\,j=1,\, \ldots ,\,h$$\end{document} (i≠j\documentclass[12pt]{minimal}
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\begin{document}$$i\ne j$$\end{document}). In this paper we consider the case G is the cyclic group Zn\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {Z}}_n$$\end{document} and determine the values of h for which a complete decomposition of Zn\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {Z}}_n$$\end{document} of order h exists. The result is then extended to the case G is a finite abelian group. We also investigate the existence of complete decompositions of Zn\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {Z}}_n$$\end{document} where the cardinality of each set in the decomposition is a prescribed integer ≥2\documentclass[12pt]{minimal}
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\begin{document}$$\ge 2$$\end{document}. As an application, we describe a way to construct codes over a binary alphabet using a construction of a complete decomposition of cyclic groups.
