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On a categorial isomorphism between a class of Completely Regular Codes and a class of Distance Regular Graphs

  • Josep Rifà-Coma
Submitted Contributions Error Correcting Codes: Theory And Applications
Part of the Lecture Notes in Computer Science book series (LNCS, volume 508)

Abstract

In [5] we established some relations between Distance Regular Graphs and Completely-Regular Codes in order to show the non-existence of a class of Distance-Regular Graph. In [6] we introduced the concept of Propelinear Code which is the algebraic structure associated to an e-Latticed, Distance-Regular Graph.

In this paper we present a functorial isomorphism between two categories, the category of e-Latticed, Distance Regular Graphs and the category of Completely Regular Binary Propelinear Codes. Starting from this categorial isomorphism it is easy to translate some properties from one category to the other, specially the non-existence properties of certain types of graphs (see 5.1).

We also show in this paper that the linear structure is the only possible propelinear structure in Golay Codes G23 and G24. From this fact we deduce the uniqueness of certain types of graphs given by its parameters (see 5.2 and cf. [2] Chap.11).

This paper was originally conceived to try to answer a question asked by Prof. Brouwer (see theorem 11).

The categorial treatment owes itself to talks that I have had with Prof. O. Moreno.

Keywords

Undirected Graph Intersection Matrix Linear Code Code Word Graph Isomorphism 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    E. Bannai, T. Ito, Algebraic Combinatorics I. (The Benjamin-Cummings Publishing Co.,Inc., California, 1984).Google Scholar
  2. [2]
    A.Brouwer, A.M.Cohen, A.Neumaier, Distance Regular Graphs. (Springer-Verlag, 1989).Google Scholar
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    P.Delsarte, An algebraic approach to the association schemes of coding theory, Philips Research Reps., Suppl. 10, (1973).Google Scholar
  4. [4]
    F.J.Macwilliams, N.J.A.Sloane, The Theory of Error-Correcting Codes, (North-Holland Publishing Company, 1977).Google Scholar
  5. [5]
    J. Rifà, L. Huguet, Classification of a Class of Distance-Regular Graphs via Completely Regular Codes. Discrete Applied Mathematics 26 (1990) 289–300.Google Scholar
  6. [6]
    J. Rifà, Distance Regular Graphs and Propelinear Codes. Springer-Verlag, LNCS 357, (1989) 341–355.Google Scholar
  7. [7]
    H.C.J. Van Tilborg, Uniformly Packed Codes, Doctoral Dissertation, Eindhoven University, (1976).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Josep Rifà-Coma
    • 1
  1. 1.Dpt. d'InformàticaUniversitat Autònoma de BarcelonaCataloniaSpain

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