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For most discrete memoryless channels, there does not exist a linear code forthe channel which uses all of the channel's input symbols. Therefore, linearityof the code for such channels is a very restrictive condition and there shouldbe a loosening of the algebraic structure of the code to a degree that the codecan admit any channel input alphabet. For any channel input alphabet size,there always exists an Abelian group structure defined on the alphabet. Weinvestigate the capacity of Abelian group codes over discrete memorylesschannels and provide lower and upper bounds on the capacity.

On the Capacity of Abelian Group Codes Over Discrete Memoryless Channels