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Abstract: In this work, the cardinality of the minimal -covers of finite rings with respect to the RT-metric is established. By generalizing the result in Nakaoka and dos Santos (2010) , the minimal cardinalities of 0-short coverings of finite chain rings are calculated. The connection between -short coverings of rings with respect to the RT-metric and the 0-short coverings of rings is demonstrated, and with the help of this connection, the problem of finding the minimal cardinalities of -short coverings of finite chain rings is solved. [Copyright &y& Elsevier]

Abstract: It is a well established fact that -spotty byte error control codes provide a good source for detecting and correcting errors in semiconductor memory systems using high-density RAM chips with wide I/O data (e.g. 8, 16 or 32 bits). Recently, a MacWilliams identity that establishes an important relation between an -spotty weight enumerator of a binary code and its dual has been proven in Kazuyoshi Suzuki et al. (2007) . In this paper, we introduce the -spotty Lee weights and the -spotty Lee weight enumerator of a quaternary code and prove a MacWilliams type identity. [Copyright &y& Elsevier]