An Introduction to Ring-Linear Coding Theory

Marcus Greferath ()
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Marcus Greferath: University College Dublin, School of Mathematical Sciences

A chapter in Gröbner Bases, Coding, and Cryptography, 2009, pp 219-238 from Springer

Abstract: Abstract This contribution gives an introduction to algebraic coding theory over rings. We will start with a historical sketch and then present basics on rings and modules. Particular attention will be paid to weight functions on these, before some foundational results of ring-linear coding will be discussed. Among these we will deal with code equivalence, and with MacWilliams’ identities about the relation between weight enumerators. A further section is devoted to existence bounds and code optimality. An outlook will then be presented on the still unsolved problem of the construction of large families of ring-linear codes of high quality.

Keywords: Linear Code; Cyclic Code; Chain Ring; Weight Enumerator; Galois Ring (search for similar items in EconPapers)
Date: 2009
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-540-93806-4_13

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DOI: 10.1007/978-3-540-93806-4_13

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