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On the classification of perfect codes: Extended side class structures
Department of Mathematics, KTH, Stockholm, Sweden.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
Department of Mathematics, KTH, Stockholm, Sweden.
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 π of the set of the n coordinate positions and a word such that . Hessler defined C and C to be linearly equivalent if there exists a non-singular linear map φ such that C=φ(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 φ and a word such that

Heden and Hessler, associated with each linear equivalence class an invariant LC 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 LC.

This fact will be used to give, in some particular cases, a complete enumeration of all extended equivalence classes of perfect codes.

Place, publisher, year, edition, pages
Amsterdam, Netherlands: Elsevier, 2010. Vol. 310, no 1, 43-55 p.
Keyword [en]
Perfect codes, Side class structures
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-51719DOI: 10.1016/j.disc.2009.07.023ISI: 000272437800007OAI: oai:DiVA.org:liu-51719DiVA: diva2:277119
Available from: 2009-11-16 Created: 2009-11-16 Last updated: 2017-12-12Bibliographically approved
In thesis
1. Optimization, Matroids and Error-Correcting Codes
Open this publication in new window or tab >>Optimization, Matroids and Error-Correcting Codes
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The first subject we investigate in this thesis deals with optimization problems on graphs. The edges are given costs defined by the values of independent exponential random variables. We show how to calculate some or all moments of the distributions of the costs of some optimization problems on graphs.

The second subject that we investigate is 1-error correcting perfect binary codes, perfect codes for short. In most work about perfect codes, two codes are considered equivalent if there is an isometric mapping between them. We call this isometric equivalence. Another type of equivalence is given if two codes can be mapped on each other using a non-singular linear map. We call this linear equivalence. A third type of equivalence is given if two codes can be mapped on each other using a composition of an isometric map and a non-singular linear map. We call this extended equivalence.

  • In Paper 1 we give a new better bound on how much the cost of the matching problem with exponential edge costs varies from its mean.
  • In Paper 2 we calculate the expected cost of an LP-relaxed version of the matching problem where some edges are given zero cost. A special case is when the vertices with probability 1 – p have a zero cost loop, for this problem we prove that the expected cost is given by a formula.
  • In Paper 3 we define the polymatroid assignment problem and give a formula for calculating all moments of its cost.
  • In Paper 4 we present a computer enumeration of the 197 isometric equivalence classes of the perfect codes of length 31 of rank 27 and with a kernel of dimension 24.
  • In Paper 5 we investigate when it is possible to map two perfect codes on each other using a non-singular linear map.
  • In Paper 6 we give an invariant for the equivalence classes of all perfect codes of all lengths when linear equivalence is considered.
  • In Paper 7 we give an invariant for the equivalence classes of all perfect codes of all lengths when extended equivalence is considered.
  • In Paper 8 we define a class of perfect codes that we call FRH-codes. It is shown that each FRH-code is linearly equivalent to a so called Phelps code and that this class contains Phelps codes as a proper subset.
Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2009. 55 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1277
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-51722 (URN)978-91-7393-521-0 (ISBN)
Public defence
2009-12-16, Nobel (BL32), hus B, ing. 23, Campus Valla, Linköpings universitet, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2009-11-16 Created: 2009-11-16 Last updated: 2010-01-12Bibliographically approved

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Hessler, Martin

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