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Parity check systems, perfect codes and codes over Frobenius rings
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). (Kombinatorik)
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: urn:nbn:se:kth:diva-67336ISBN: 978-91-7501-237-7 (print)OAI: oai:DiVA.org:kth-67336DiVA: diva2:484869
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
List of papers
1. Non Phelps codes of length 15 and rank 14
Open this publication in new window or tab >>Non Phelps codes of length 15 and rank 14
2007 (English)In: The Australasian Journal of Combinatorics, ISSN 1034-4942, Vol. 38, 141-153 p.Article in journal (Refereed) Published
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-71313 (URN)2-s2.0-70349340864 (Scopus ID)
Note
QC 20120131Available from: 2012-01-31 Created: 2012-01-31 Last updated: 2017-12-08Bibliographically approved
2. On the classification of perfect codes: Extended side class structures
Open this publication in new window or tab >>On the classification of perfect codes: Extended side class structures
2010 (English)In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 310, no 1, 43-55 p.Article in journal (Refereed) Published
Abstract [en]

The two 1-error correcting perfect binary codes. C and C' are said to be equivalent if there exists a permutation pi of the set of the n coordinate positions and a word (d) over bar such that C' = pi((d) over bar + C). Hessler defined C and C' to be linearly equivalent if there exists a non-singular linear map phi such that C' = phi(C). Two perfect codes C and C' of length n will be defined to be extended equivalent if there exists a non-singular linear map W and a word (d) over bar such that C' = phi((d) over bar + C). Heden and Hessler, associated with each linear equivalence class an invariant L-C and this invariant was shown to be a subspace of the kernel of some perfect code. It is shown here that, in the case of extended equivalence, the corresponding invariant will be the extension of the code L-C. This fact will be used to give, in some particular cases, a complete enumeration of all extended equivalence classes of perfect codes.

Keyword
Perfect codes, Side class structures, rank
Identifiers
urn:nbn:se:kth:diva-19017 (URN)10.1016/j.disc.2009.07.023 (DOI)000272437800007 ()2-s2.0-70350716446 (Scopus ID)
Note
QC 20100525Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2012-01-31Bibliographically approved
3. ON THE EXISTENCE OF EXTENDED PERFECT BINARY CODES WITH TRIVIAL SYMMETRY GROUP
Open this publication in new window or tab >>ON THE EXISTENCE OF EXTENDED PERFECT BINARY CODES WITH TRIVIAL SYMMETRY GROUP
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.

Keyword
Perfect codes, symmetry group, error-correcting codes, construction
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-18702 (URN)10.3934/amc.2009.3.295 (DOI)000269220600007 ()2-s2.0-69249097938 (Scopus ID)
Note
QC 20100525Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2012-02-01Bibliographically approved
4. on the symmetry group of extended perfect binary codes of length n+1 and rank n-log(n+1)+2
Open this publication in new window or tab >>on the symmetry group of extended perfect binary codes of length n+1 and rank n-log(n+1)+2
2012 (English)In: Advances in Mathematics of Communication, ISSN 1930-5346, Vol. 6, no 2, 121-130 p.Article in journal (Refereed) Published
Abstract [en]

It is proved that for every integer n = 2(k) - 1, with k >= 5, there exists a perfect code C of length n, of rank r = n - log(n + 1) + 2 and with a trivial symmetry group. This result extends an earlier result by the authors that says that for any length n = 2(k) - 1, with k >= 5, and any rank r, with n - log(n + 1) + 3 <= r <= n - 1 there exist perfect codes with a trivial symmetry group.

Keyword
Perfect codes, symmetry group
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-71321 (URN)10.3934/amc.2012.6.121 (DOI)000304194500001 ()2-s2.0-84861967067 (Scopus ID)
Funder
Knut and Alice Wallenberg Foundation, KAW 2005.0098
Note
QC 20120612Available from: 2012-01-31 Created: 2012-01-31 Last updated: 2012-06-12Bibliographically approved
5. Parity check systems of codes over finite commutative Frobeniusrings and finite Abelian groups
Open this publication in new window or tab >>Parity check systems of codes over finite commutative Frobeniusrings and finite Abelian groups
(English)Article in journal (Other academic) Submitted
National Category
Algebra and Logic
Identifiers
urn:nbn:se:kth:diva-71324 (URN)
Note
QS 20120328Available from: 2012-01-31 Created: 2012-01-31 Last updated: 2012-03-28Bibliographically approved

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