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In this work we focus on a connection between sumsets and covering codes in an arbitrary finite module. For this purpose, bounds on a new problem on sumsets are obtained from well-known results of additive number theory, namely, the Cauchy-Davenport theorem, the Vosper theorem and a theorem due to Hamidoune-Rødseth. As an application, the approach is able to extend the Blokhuis-Lam theorems and a construction of covering codes by Honkala to an arbitrary module.

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