Abstract
Foundational and theoretical aspects of algebraic coding theory are discussed with the concentration in the classes of cyclic and negacyclic codes over finite chain rings. The significant role of finite rings as alphabets in coding theory is presented. We surveys results on both simple-root and repeated-root cases of such codes. Many directions in which the notions of cyclicity and negacyclicity have been generalized are also considered. The paper is devoted to giving an introduction to this area of applied algebra. We do not intend to be encyclopedic, the topics included are bounded to reflect our own research interest.