A systematic group code is a code over of order defined by homomorphisms which determine the parity check bits. The remaining bits are the information bits themselves.
Group codes can be constructed by special generator matrices which resemble generator matrices of linear block codes except that the elements of those matrices are endomorphisms of the group instead of symbols from the code's alphabet. For example, consider the generator matrix
The elements of this matrix are matrices which are endomorphisms. In this scenario, each codeword can be represented as where are the generators of .
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