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Parity check systems, perfect codes and codes over Frobenius ringsPrimeFaces.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}); 2012 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

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

Stockholm: KTH Royal Institute of Technology, 2012. , vii, 28 p.
##### Series

Trita-MAT. MA, ISSN 1401-2278 ; 11:12
##### National Category

Mathematics
##### Identifiers

URN: urn:nbn:se:kth:diva-67336ISBN: 978-91-7501-237-7OAI: oai:DiVA.org:kth-67336DiVA: diva2:484869
##### Public defence

2012-02-17, Sal F3, Lindstedtsvägen 26, KTH, Stockholm, 13:00 (English)
##### Opponent

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##### Supervisors

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#####

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##### Note

QC 20120131Available from: 2012-01-31 Created: 2012-01-27 Last updated: 2012-01-31Bibliographically approved
##### List of papers

This thesis consists of five papers related to coding theory. The first four papers are mainly devoted to perfect 1-error correcting binary codes. The fifth paper concerns codes over finite Abelian groups and finite commutative Frobenius rings.

In Paper A we construct a new class of perfect binary codes of length 15. These codes can not be obtained by a construction of Phelps and Solov’eva. The verification of the existence of these kind of codes gives an answer to a question by Zinoviev and Zinoviev from 2003.

In Paper B the concept of extended equivalence for binary codes is introduced. A linear code L** _{C}*, which is an invariant for this equivalence relation, is associated with every perfect binary code

In Paper C and D we prove that there exist perfect binary codes and extended perfect binary codes with a trivial symmetry group for most admissible cases of lengths and ranks. The results of these two papers have, together with previously known results, completely solved the problem of for which lengths and ranks there exist perfect binary codes with a trivial symmetry group, except in a handful of cases.

In Paper E the concept of parity check matrices of linear codes over finite fields is generalized to parity check systems of both linear and nonlinear codes over finite Abelian groups and finite commutative Frobenius rings. A parity check system is a concatenation of two matrices and can be found by the use of Fourier analysis over finite Abelian groups. It is shown how some fundamental properties of a code can be derived from the set of columns or the set of rows in an associated parity check system. Furthermore, in Paper E, Cayley graphs and integral group rings are associated with parity check systems in order to investigate some problems in coding theory.

1. Non Phelps codes of length 15 and rank 14$(function(){PrimeFaces.cw("OverlayPanel","overlay486644",{id:"formSmash:j_idt503:0:j_idt507",widgetVar:"overlay486644",target:"formSmash:j_idt503:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. On the classification of perfect codes: Extended side class structures$(function(){PrimeFaces.cw("OverlayPanel","overlay337064",{id:"formSmash:j_idt503:1:j_idt507",widgetVar:"overlay337064",target:"formSmash:j_idt503:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. ON THE EXISTENCE OF EXTENDED PERFECT BINARY CODES WITH TRIVIAL SYMMETRY GROUP$(function(){PrimeFaces.cw("OverlayPanel","overlay336749",{id:"formSmash:j_idt503:2:j_idt507",widgetVar:"overlay336749",target:"formSmash:j_idt503:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

4. on the symmetry group of extended perfect binary codes of length n+1 and rank n-log(n+1)+2$(function(){PrimeFaces.cw("OverlayPanel","overlay486653",{id:"formSmash:j_idt503:3:j_idt507",widgetVar:"overlay486653",target:"formSmash:j_idt503:3:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

5. Parity check systems of codes over finite commutative Frobeniusrings and finite Abelian groups$(function(){PrimeFaces.cw("OverlayPanel","overlay486722",{id:"formSmash:j_idt503:4:j_idt507",widgetVar:"overlay486722",target:"formSmash:j_idt503:4:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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