On the role of rings and modules in algebraic coding theory

Foundational and theoretical aspects of coding theory over rings and modules are considered. Topics discussed include the role of rings and modules as alphabets in coding theory and the necessity of exploring metrics other than the traditional Hamming weight on such alphabets. We survey attempts to extend the classical results of coding theory over finite fields to this new setting and consider, in particular, the extension of the MacWilliams equivalence theorem and of the MacWilliams duality identities to the context of codes over finite Frobenius rings. We also discuss arguments that seem to point at finite Frobenius rings or to character modules over arbitrary rings as the natural alphabets for an extended coding theory that would still include such theoretical staples. Work on the establishment of bounds on the parameters of ring-linear codes is also sampled as are constructions of codes and the design of decoding algorithms.