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Abstract: Cellular automata are used to model dynamical phenomena by focusing on their local behavior which depends on the neighboring cells in order to express their global behavior. The geometrical structure of the models suggests the algebraic structure of cellular automata. After modeling the dynamical phenomena, it is sometimes an important problem to be able to move backwards in order to understand it better. This is only possible if cellular automata is reversible. In this paper, 2D finite cellular automata defined by local rules based on hexagonal cell structure are studied. Rule matrix of the hexagonal finite cellular automaton is obtained. The rank of rule matrices representing the 2D hexagonal finite cellular automata via an algorithm is computed. It is a well known fact that determining the reversibility of a 2D cellular automata is a very difficult problem in general. Here, the reversibility problem of this family of 2D hexagonal cellular automata is also resolved completely. [Copyright &y& Elsevier]

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]