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978-3-8439-2165-7, Reihe Mathematik
Gröbner Bases in Coding Theory
191 Seiten, Dissertation Technische Universität Hamburg-Harburg (2015), Softcover, B5
Coding theory is a relatively young subject of mathematics and computer science which, however, gained immensely in importance in the course of the digital age. Its main objective is the construction and study of schemes for the reliable transmission of data over noisy channels. Linear codes being an important class of block codes are instances of such schemes.
Gröbner bases, on the other hand, have their origin in commutative algebra. They form generating sets for ideals in the multivariate polynomial ring which allow for the generalization of many well-known results in the univariate case.
In this thesis Gröbner bases are employed for the study of linear codes. The foundation of this work is an association of linear codes with binomial ideals in the multivariate polynomial ring. One advantage of this connection is that it yields a uniform framework applicable to all linear codes.
Specifically, the following contributions are made in this thesis: Connections between binomial ideals associated to linear codes and toric ideals are provided. In this way, an adaptation of a Gröbner bases based algorithm for computing Hilbert bases results in a new method for computing the kernel of a matrix over a finite field. Moreover, Gröbner structures for linear codes are studied. It is shown that lexicographic Gröbner bases can be read off from generator matrices in standard form thus revealing a deep connection between the information sets and the lexicographic Gröbner bases of a linear code.
Lawrence liftings are generalized and as a result a procedure for computing the Graver basis of a code is developed. Following this, a method that computes the universal Gröbner basis of a code from its Graver basis is given. Special cases such as binary codes, codes over finite fields of characteristic 2 and perfect codes are further examined. Additionally, an efficient method that computes the part of the Gröbner fan for a linear code that consists exactly of all degree compatible Gröbner bases is established.
To conclude, several applications of Gröbner structures in coding theory are presented. For instance, a novel heuristic decoding algorithm based on a degree compatible Gröbner basis is provided.