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An Introduction to Ring-Linear Coding Theory

In: Gröbner Bases, Coding, and Cryptography

Author

Listed:
  • Marcus Greferath

    (University College Dublin, School of Mathematical Sciences)

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.

Suggested Citation

  • Marcus Greferath, 2009. "An Introduction to Ring-Linear Coding Theory," Springer Books, in: Massimiliano Sala & Shojiro Sakata & Teo Mora & Carlo Traverso & Ludovic Perret (ed.), Gröbner Bases, Coding, and Cryptography, pages 219-238, Springer.
    • Handle: RePEc:spr:sprchp:978-3-540-93806-4_13
    DOI: 10.1007/978-3-540-93806-4_13
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