On the role of rings and modules in algebraic coding theory

Marcus Greferath, Sergio R. López-Permouth

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Abstract

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.
Original languageEnglish
Title of host publicationGroups, rings and group rings
EditorsAntonio Giambruno
PublisherChapman and Hall
Pages205-216
Number of pages12
Volume248
ISBN (Electronic)978-1-4200-1096-1
ISBN (Print)978-1-58488-581-8
DOIs
Publication statusPublished - 2006
MoE publication typeNot Eligible

Publication series

NameLecture Notes in Pure and Applied Mathematics
PublisherChapman Hall/CRC, Boca Raton, FL

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