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ON THE EXISTENCE OF EXTENDED PERFECT BINARY CODES WITH TRIVIAL SYMMETRY GROUP
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2009 (English)In: Advances in Mathematics of Communications, ISSN 1930-5346, Vol. 3, no 3, 295-309 p.Article in journal (Refereed) Published
Abstract [en]

The set of permutations of the coordinate set that maps a perfect code C into itself is called the symmetry group of C and is denoted by Sym(C). It is proved that for all integers n = 2(m) - 1, where m = 4, 5, 6, ... , and for any integer r, where n - log(n + 1) + 3 <= r <= n - 1, there are perfect codes of length n and rank r with a trivial symmetry group, i.e. Sym(C) = {id}. The result is shown to be true, more generally, for the extended perfect codes of length n + 1.

Place, publisher, year, edition, pages
2009. Vol. 3, no 3, 295-309 p.
Keyword [en]
Perfect codes, symmetry group, error-correcting codes, construction
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-18702DOI: 10.3934/amc.2009.3.295ISI: 000269220600007Scopus ID: 2-s2.0-69249097938OAI: oai:DiVA.org:kth-18702DiVA: diva2:336749
Note
QC 20100525Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2012-02-01Bibliographically approved
In thesis
1. Parity check systems, perfect codes and codes over Frobenius rings
Open this publication in new window or tab >>Parity check systems, perfect codes and codes over Frobenius rings
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

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 C. By using L*C we give, in some particular cases, a complete enumeration of the extended equivalence classes of perfect binary codes.

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.

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:nbn:se:kth:diva-67336 (URN)978-91-7501-237-7 (ISBN)
Public defence
2012-02-17, Sal F3, Lindstedtsvägen 26, KTH, Stockholm, 13:00 (English)
Opponent
Supervisors
Note
QC 20120131Available from: 2012-01-31 Created: 2012-01-27 Last updated: 2012-01-31Bibliographically approved

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