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On the classification of perfect codes: Side class structures
Department of Mathematics, KTH, S-100 44 Stockholm, Sweden.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
2006 (English)In: Designs, Codes and Cryptography, ISSN 0925-1022, E-ISSN 1573-7586, Vol. 40, no 3, 319-333 p.Article in journal (Refereed) Published
Abstract [en]

The side class structure of a perfect 1-error correcting binary code (hereafter referred to as a perfect code) C describes the linear relations between the coset representatives of the kernel of C. Two perfect codes C and C' are linearly equivalent if there exists a non-singular matrix A such that AC = C' where C and C' are matrices with the code words of C and C' as columns. Hessler proved that the perfect codes C and C' are linearly equivalent if and only if they have isomorphic side class structures. The aim of this paper is to describe all side class structures. It is shown that the transpose of any side class structure is the dual of a subspace of the kernel of some perfect code and vice versa, any dual of a subspace of a kernel of some perfect code is the transpose of the side class structure of some perfect code. The conclusion is that for classification purposes of perfect codes it is sufficient to find the family of all kernels of perfect codes. © Springer Science + Business Media, LLC 2006.

Place, publisher, year, edition, pages
2006. Vol. 40, no 3, 319-333 p.
Keyword [en]
Perfect codes, Tilings
National Category
Engineering and Technology
Identifiers
URN: urn:nbn:se:liu:diva-50133DOI: 10.1007/s10623-006-0024-4OAI: oai:DiVA.org:liu-50133DiVA: diva2:271029
Available from: 2009-10-11 Created: 2009-10-11 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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