Abstract: Described are a system and method for transmitting digital messages through wire channels and telecommunication channels using electromagnetic waves. The use of the channels is simplified by excluding multiplication and division operators from the coding and a decoding process. The system and method allows to transmit any messages from elements of Abelian group including code words whose elements are matrixes, polynomials, numbers of mixed-base notation and nonpositional notation. An encoder of the system may include a driver clock, a function g<2> calculator, a pulse generator having recurrent frequency of f(k+1)/k, a pulse recurrent frequency doubler, a ring counter up to k, an adder-accumulator of elements of Abelian group, a key, a ring counter up to (2k+1), an AND component, a main memory unit, a key, a trigger, a main memory unit and a ring counter up to (k+1).