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A Gröbner basis algorithm for fast encoding of Reed-Müller codesPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2016 (English)Independent thesis Basic level (degree of Bachelor), 10,5 credits / 16 HE creditsStudent thesis
##### Abstract [en]

##### Place, publisher, year, edition, pages

2016. , 35 p.
##### Series

LiTH-MAT-EX, 2016/06
##### Keyword [en]

Gröbner basis, error correcting codes, coding theory, algebra, Reed-Müller
##### Keyword [sv]

Gröbnerbas, felrättande koder, kodningsteori, algebra, Reed-Müller
##### National Category

Algebra and Logic
##### Identifiers

URN: urn:nbn:se:liu:diva-132429ISRN: LiTH-MAT-EX–2016/06–SEOAI: oai:DiVA.org:liu-132429DiVA: diva2:1045846
##### Subject / course

Mathematics
#####

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##### Supervisors

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt406",{id:"formSmash:j_idt406",widgetVar:"widget_formSmash_j_idt406",multiple:true});
##### Examiners

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt412",{id:"formSmash:j_idt412",widgetVar:"widget_formSmash_j_idt412",multiple:true});
Available from: 2016-11-16 Created: 2016-11-10 Last updated: 2016-11-16Bibliographically approved

In this thesis the relationship between Gröbner bases and algebraic coding theory is investigated, and especially applications towards linear codes, with Reed-Müller codes as an illustrative example. We prove that each linear code can be described as a binomial ideal of a polynomial ring, and that a systematic encoding algorithm for such codes is given by the remainder of the information word computed with respect to the reduced Gröbner basis. Finally we show how to apply the representation of a code by its corresponding polynomial ring ideal to construct a class of codes containing the so called primitive Reed-Müller codes, with a few examples of this result.

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