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Coding theory via Groebner bases
Citation Link: https://doi.org/10.15480/882.1081
Other Titles
Codierungstheorie mittels Gröbnerbasen
Publikationstyp
Doctoral Thesis
Date Issued
2012
Sprache
English
Author(s)
Advisor
Title Granting Institution
Technische Universität Hamburg
Place of Title Granting Institution
Hamburg
Examination Date
2012-02-14
Institut
TORE-DOI
Coding theory plays an important role in efficient transmission of data over noisy channels.In this thesis efficient encoding procedure for linear codes is developed using an algebraic approach. Description of linear codes as ideals in a residue class ring are given in terms of Groebner basis. While investigating primitive Reed Muller codes, a special family of linear codes with designed Hamming distance is obtained. A result proves their superiority over existing primitive Reed Muller codes. Furthermore, codes associated to a particular binomial ideal, defined as a sum of toric ideal and a prime ideal, are explored through minimal generators and Groebner basis. For these non-toric binomial ideals universal Groebner bases, Graver bases and circuits are also found. It is shown that each such binomial ideal has a natural reduced Groebner basis which provides a very compact encoding procedure. Finally, the binomial ideal of a linear code is presented in terms of its syzygy modules and the corresponding finite free resolution is also given.
Subjects
Gröbnerbase
linearer Code
binomiales Ideal
Syzygie
Groebner base
linear code, binomial ideal, syzygy
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