Cyclic codes over rings

Abstract

Coding theory originated as the mathematical foundation for the transmission of messages over noisy communication channels and deals with the problem of detecting and correcting transmission errors caused by the noise of the channel. The mathematical background needed for example was linear algebra, theory of groups, rings and nite elds, and other areas of discrete mathematics, as theory of designs. Thus, coding theory has now become and active part of mathematical research. On the family of codes, linear codes are special codes with rich mathematical structure. For this reason these codes are easy to construct, encoding and decoding linear codes is very quick and easy by the linearity of a code. One of the most studied class of linear codes are cyclic codes. They can be eciently implemented and they have a rich algebraic structure, for this reason many practically important codes are cyclic.. Coding theory originated as the mathematical foundation for the transmissionof messages over noisy communication channels and deals with the problem ofdetecting and correcting transmission errors caused by the noise of thechannel.On the family of codes, linear codes are special codes with rich mathematicalstructure.One ofthe most studied class of linear codes are cyclic codes. They can bee cientlyimplemented and they have a rich algebraic structure, for this reason manypractically important codes are cyclic.